// The following is adapted from fdlibm (http://www.netlib.org/fdlibm).
//
// ====================================================
// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
//
// Developed at SunSoft, a Sun Microsystems, Inc. business.
// Permission to use, copy, modify, and distribute this
// software is freely granted, provided that this notice
// is preserved.
// ====================================================
//
// The original source code covered by the above license above has been
// modified significantly by Google Inc.
// Copyright 2016 the V8 project authors. All rights reserved.

#include "src/base/ieee754.h"

#include <cmath>
#include <limits>

#include "src/base/build_config.h"
#include "src/base/macros.h"
#include "src/base/overflowing-math.h"

namespace v8 {
namespace base {
    namespace ieee754 {

        namespace {

            /* Disable "potential divide by 0" warning in Visual Studio compiler. */

#if V8_CC_MSVC

#pragma warning(disable : 4723)

#endif

            /*
 * The original fdlibm code used statements like:
 *  n0 = ((*(int*)&one)>>29)^1;   * index of high word *
 *  ix0 = *(n0+(int*)&x);     * high word of x *
 *  ix1 = *((1-n0)+(int*)&x);   * low word of x *
 * to dig two 32 bit words out of the 64 bit IEEE floating point
 * value.  That is non-ANSI, and, moreover, the gcc instruction
 * scheduler gets it wrong.  We instead use the following macros.
 * Unlike the original code, we determine the endianness at compile
 * time, not at run time; I don't see much benefit to selecting
 * endianness at run time.
 */

            /*
 * A union which permits us to convert between a double and two 32 bit
 * ints.
 * TODO(jkummerow): This is undefined behavior. Use bit_cast instead.
 */

#if V8_TARGET_LITTLE_ENDIAN

            union ieee_double_shape_type {
                double value;
                struct {
                    uint32_t lsw;
                    uint32_t msw;
                } parts;
                struct {
                    uint64_t w;
                } xparts;
            };

#else

            union ieee_double_shape_type {
                double value;
                struct {
                    uint32_t msw;
                    uint32_t lsw;
                } parts;
                struct {
                    uint64_t w;
                } xparts;
            };

#endif

            /* Get two 32 bit ints from a double.  */

#define EXTRACT_WORDS(ix0, ix1, d)   \
    do {                             \
        ieee_double_shape_type ew_u; \
        ew_u.value = (d);            \
        (ix0) = ew_u.parts.msw;      \
        (ix1) = ew_u.parts.lsw;      \
    } while (false)

/* Get a 64-bit int from a double. */
#define EXTRACT_WORD64(ix, d)        \
    do {                             \
        ieee_double_shape_type ew_u; \
        ew_u.value = (d);            \
        (ix) = ew_u.xparts.w;        \
    } while (false)

            /* Get the more significant 32 bit int from a double.  */

#define GET_HIGH_WORD(i, d)          \
    do {                             \
        ieee_double_shape_type gh_u; \
        gh_u.value = (d);            \
        (i) = gh_u.parts.msw;        \
    } while (false)

            /* Get the less significant 32 bit int from a double.  */

#define GET_LOW_WORD(i, d)           \
    do {                             \
        ieee_double_shape_type gl_u; \
        gl_u.value = (d);            \
        (i) = gl_u.parts.lsw;        \
    } while (false)

            /* Set a double from two 32 bit ints.  */

#define INSERT_WORDS(d, ix0, ix1)    \
    do {                             \
        ieee_double_shape_type iw_u; \
        iw_u.parts.msw = (ix0);      \
        iw_u.parts.lsw = (ix1);      \
        (d) = iw_u.value;            \
    } while (false)

/* Set a double from a 64-bit int. */
#define INSERT_WORD64(d, ix)         \
    do {                             \
        ieee_double_shape_type iw_u; \
        iw_u.xparts.w = (ix);        \
        (d) = iw_u.value;            \
    } while (false)

            /* Set the more significant 32 bits of a double from an int.  */

#define SET_HIGH_WORD(d, v)          \
    do {                             \
        ieee_double_shape_type sh_u; \
        sh_u.value = (d);            \
        sh_u.parts.msw = (v);        \
        (d) = sh_u.value;            \
    } while (false)

            /* Set the less significant 32 bits of a double from an int.  */

#define SET_LOW_WORD(d, v)           \
    do {                             \
        ieee_double_shape_type sl_u; \
        sl_u.value = (d);            \
        sl_u.parts.lsw = (v);        \
        (d) = sl_u.value;            \
    } while (false)

            /* Support macro. */

#define STRICT_ASSIGN(type, lval, rval) ((lval) = (rval))

            int32_t __ieee754_rem_pio2(double x, double* y) V8_WARN_UNUSED_RESULT;
            double __kernel_cos(double x, double y) V8_WARN_UNUSED_RESULT;
            int __kernel_rem_pio2(double* x, double* y, int e0, int nx, int prec,
                const int32_t* ipio2) V8_WARN_UNUSED_RESULT;
            double __kernel_sin(double x, double y, int iy) V8_WARN_UNUSED_RESULT;

            /* __ieee754_rem_pio2(x,y)
 *
 * return the remainder of x rem pi/2 in y[0]+y[1]
 * use __kernel_rem_pio2()
 */
            int32_t __ieee754_rem_pio2(double x, double* y)
            {
                /*
   * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
   */
                static const int32_t two_over_pi[] = {
                    0xA2F983,
                    0x6E4E44,
                    0x1529FC,
                    0x2757D1,
                    0xF534DD,
                    0xC0DB62,
                    0x95993C,
                    0x439041,
                    0xFE5163,
                    0xABDEBB,
                    0xC561B7,
                    0x246E3A,
                    0x424DD2,
                    0xE00649,
                    0x2EEA09,
                    0xD1921C,
                    0xFE1DEB,
                    0x1CB129,
                    0xA73EE8,
                    0x8235F5,
                    0x2EBB44,
                    0x84E99C,
                    0x7026B4,
                    0x5F7E41,
                    0x3991D6,
                    0x398353,
                    0x39F49C,
                    0x845F8B,
                    0xBDF928,
                    0x3B1FF8,
                    0x97FFDE,
                    0x05980F,
                    0xEF2F11,
                    0x8B5A0A,
                    0x6D1F6D,
                    0x367ECF,
                    0x27CB09,
                    0xB74F46,
                    0x3F669E,
                    0x5FEA2D,
                    0x7527BA,
                    0xC7EBE5,
                    0xF17B3D,
                    0x0739F7,
                    0x8A5292,
                    0xEA6BFB,
                    0x5FB11F,
                    0x8D5D08,
                    0x560330,
                    0x46FC7B,
                    0x6BABF0,
                    0xCFBC20,
                    0x9AF436,
                    0x1DA9E3,
                    0x91615E,
                    0xE61B08,
                    0x659985,
                    0x5F14A0,
                    0x68408D,
                    0xFFD880,
                    0x4D7327,
                    0x310606,
                    0x1556CA,
                    0x73A8C9,
                    0x60E27B,
                    0xC08C6B,
                };

                static const int32_t npio2_hw[] = {
                    0x3FF921FB,
                    0x400921FB,
                    0x4012D97C,
                    0x401921FB,
                    0x401F6A7A,
                    0x4022D97C,
                    0x4025FDBB,
                    0x402921FB,
                    0x402C463A,
                    0x402F6A7A,
                    0x4031475C,
                    0x4032D97C,
                    0x40346B9C,
                    0x4035FDBB,
                    0x40378FDB,
                    0x403921FB,
                    0x403AB41B,
                    0x403C463A,
                    0x403DD85A,
                    0x403F6A7A,
                    0x40407E4C,
                    0x4041475C,
                    0x4042106C,
                    0x4042D97C,
                    0x4043A28C,
                    0x40446B9C,
                    0x404534AC,
                    0x4045FDBB,
                    0x4046C6CB,
                    0x40478FDB,
                    0x404858EB,
                    0x404921FB,
                };

                /*
   * invpio2:  53 bits of 2/pi
   * pio2_1:   first  33 bit of pi/2
   * pio2_1t:  pi/2 - pio2_1
   * pio2_2:   second 33 bit of pi/2
   * pio2_2t:  pi/2 - (pio2_1+pio2_2)
   * pio2_3:   third  33 bit of pi/2
   * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)
   */

                static const double
                    zero
                    = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
                    half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
                    two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
                    invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
                    pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
                    pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
                    pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
                    pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
                    pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
                    pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */

                double z, w, t, r, fn;
                double tx[3];
                int32_t e0, i, j, nx, n, ix, hx;
                uint32_t low;

                z = 0;
                GET_HIGH_WORD(hx, x); /* high word of x */
                ix = hx & 0x7FFFFFFF;
                if (ix <= 0x3FE921FB) { /* |x| ~<= pi/4 , no need for reduction */
                    y[0] = x;
                    y[1] = 0;
                    return 0;
                }
                if (ix < 0x4002D97C) { /* |x| < 3pi/4, special case with n=+-1 */
                    if (hx > 0) {
                        z = x - pio2_1;
                        if (ix != 0x3FF921FB) { /* 33+53 bit pi is good enough */
                            y[0] = z - pio2_1t;
                            y[1] = (z - y[0]) - pio2_1t;
                        } else { /* near pi/2, use 33+33+53 bit pi */
                            z -= pio2_2;
                            y[0] = z - pio2_2t;
                            y[1] = (z - y[0]) - pio2_2t;
                        }
                        return 1;
                    } else { /* negative x */
                        z = x + pio2_1;
                        if (ix != 0x3FF921FB) { /* 33+53 bit pi is good enough */
                            y[0] = z + pio2_1t;
                            y[1] = (z - y[0]) + pio2_1t;
                        } else { /* near pi/2, use 33+33+53 bit pi */
                            z += pio2_2;
                            y[0] = z + pio2_2t;
                            y[1] = (z - y[0]) + pio2_2t;
                        }
                        return -1;
                    }
                }
                if (ix <= 0x413921FB) { /* |x| ~<= 2^19*(pi/2), medium size */
                    t = fabs(x);
                    n = static_cast<int32_t>(t * invpio2 + half);
                    fn = static_cast<double>(n);
                    r = t - fn * pio2_1;
                    w = fn * pio2_1t; /* 1st round good to 85 bit */
                    if (n < 32 && ix != npio2_hw[n - 1]) {
                        y[0] = r - w; /* quick check no cancellation */
                    } else {
                        uint32_t high;
                        j = ix >> 20;
                        y[0] = r - w;
                        GET_HIGH_WORD(high, y[0]);
                        i = j - ((high >> 20) & 0x7FF);
                        if (i > 16) { /* 2nd iteration needed, good to 118 */
                            t = r;
                            w = fn * pio2_2;
                            r = t - w;
                            w = fn * pio2_2t - ((t - r) - w);
                            y[0] = r - w;
                            GET_HIGH_WORD(high, y[0]);
                            i = j - ((high >> 20) & 0x7FF);
                            if (i > 49) { /* 3rd iteration need, 151 bits acc */
                                t = r; /* will cover all possible cases */
                                w = fn * pio2_3;
                                r = t - w;
                                w = fn * pio2_3t - ((t - r) - w);
                                y[0] = r - w;
                            }
                        }
                    }
                    y[1] = (r - y[0]) - w;
                    if (hx < 0) {
                        y[0] = -y[0];
                        y[1] = -y[1];
                        return -n;
                    } else {
                        return n;
                    }
                }
                /*
   * all other (large) arguments
   */
                if (ix >= 0x7FF00000) { /* x is inf or NaN */
                    y[0] = y[1] = x - x;
                    return 0;
                }
                /* set z = scalbn(|x|,ilogb(x)-23) */
                GET_LOW_WORD(low, x);
                SET_LOW_WORD(z, low);
                e0 = (ix >> 20) - 1046; /* e0 = ilogb(z)-23; */
                SET_HIGH_WORD(z, ix - static_cast<int32_t>(static_cast<uint32_t>(e0) << 20));
                for (i = 0; i < 2; i++) {
                    tx[i] = static_cast<double>(static_cast<int32_t>(z));
                    z = (z - tx[i]) * two24;
                }
                tx[2] = z;
                nx = 3;
                while (tx[nx - 1] == zero)
                    nx--; /* skip zero term */
                n = __kernel_rem_pio2(tx, y, e0, nx, 2, two_over_pi);
                if (hx < 0) {
                    y[0] = -y[0];
                    y[1] = -y[1];
                    return -n;
                }
                return n;
            }

            /* __kernel_cos( x,  y )
 * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
 * Input x is assumed to be bounded by ~pi/4 in magnitude.
 * Input y is the tail of x.
 *
 * Algorithm
 *      1. Since cos(-x) = cos(x), we need only to consider positive x.
 *      2. if x < 2^-27 (hx<0x3E400000 0), return 1 with inexact if x!=0.
 *      3. cos(x) is approximated by a polynomial of degree 14 on
 *         [0,pi/4]
 *                                       4            14
 *              cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
 *         where the remez error is
 *
 *      |              2     4     6     8     10    12     14 |     -58
 *      |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  )| <= 2
 *      |                                                      |
 *
 *                     4     6     8     10    12     14
 *      4. let r = C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  , then
 *             cos(x) = 1 - x*x/2 + r
 *         since cos(x+y) ~ cos(x) - sin(x)*y
 *                        ~ cos(x) - x*y,
 *         a correction term is necessary in cos(x) and hence
 *              cos(x+y) = 1 - (x*x/2 - (r - x*y))
 *         For better accuracy when x > 0.3, let qx = |x|/4 with
 *         the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
 *         Then
 *              cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
 *         Note that 1-qx and (x*x/2-qx) is EXACT here, and the
 *         magnitude of the latter is at least a quarter of x*x/2,
 *         thus, reducing the rounding error in the subtraction.
 */
            V8_INLINE double __kernel_cos(double x, double y)
            {
                static const double
                    one
                    = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
                    C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
                    C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
                    C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
                    C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
                    C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
                    C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */

                double a, iz, z, r, qx;
                int32_t ix;
                GET_HIGH_WORD(ix, x);
                ix &= 0x7FFFFFFF; /* ix = |x|'s high word*/
                if (ix < 0x3E400000) { /* if x < 2**27 */
                    if (static_cast<int>(x) == 0)
                        return one; /* generate inexact */
                }
                z = x * x;
                r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * C6)))));
                if (ix < 0x3FD33333) { /* if |x| < 0.3 */
                    return one - (0.5 * z - (z * r - x * y));
                } else {
                    if (ix > 0x3FE90000) { /* x > 0.78125 */
                        qx = 0.28125;
                    } else {
                        INSERT_WORDS(qx, ix - 0x00200000, 0); /* x/4 */
                    }
                    iz = 0.5 * z - qx;
                    a = one - qx;
                    return a - (iz - (z * r - x * y));
                }
            }

            /* __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
 * double x[],y[]; int e0,nx,prec; int ipio2[];
 *
 * __kernel_rem_pio2 return the last three digits of N with
 *              y = x - N*pi/2
 * so that |y| < pi/2.
 *
 * The method is to compute the integer (mod 8) and fraction parts of
 * (2/pi)*x without doing the full multiplication. In general we
 * skip the part of the product that are known to be a huge integer (
 * more accurately, = 0 mod 8 ). Thus the number of operations are
 * independent of the exponent of the input.
 *
 * (2/pi) is represented by an array of 24-bit integers in ipio2[].
 *
 * Input parameters:
 *      x[]     The input value (must be positive) is broken into nx
 *              pieces of 24-bit integers in double precision format.
 *              x[i] will be the i-th 24 bit of x. The scaled exponent
 *              of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
 *              match x's up to 24 bits.
 *
 *              Example of breaking a double positive z into x[0]+x[1]+x[2]:
 *                      e0 = ilogb(z)-23
 *                      z  = scalbn(z,-e0)
 *              for i = 0,1,2
 *                      x[i] = floor(z)
 *                      z    = (z-x[i])*2**24
 *
 *
 *      y[]     output result in an array of double precision numbers.
 *              The dimension of y[] is:
 *                      24-bit  precision       1
 *                      53-bit  precision       2
 *                      64-bit  precision       2
 *                      113-bit precision       3
 *              The actual value is the sum of them. Thus for 113-bit
 *              precison, one may have to do something like:
 *
 *              long double t,w,r_head, r_tail;
 *              t = (long double)y[2] + (long double)y[1];
 *              w = (long double)y[0];
 *              r_head = t+w;
 *              r_tail = w - (r_head - t);
 *
 *      e0      The exponent of x[0]
 *
 *      nx      dimension of x[]
 *
 *      prec    an integer indicating the precision:
 *                      0       24  bits (single)
 *                      1       53  bits (double)
 *                      2       64  bits (extended)
 *                      3       113 bits (quad)
 *
 *      ipio2[]
 *              integer array, contains the (24*i)-th to (24*i+23)-th
 *              bit of 2/pi after binary point. The corresponding
 *              floating value is
 *
 *                      ipio2[i] * 2^(-24(i+1)).
 *
 * External function:
 *      double scalbn(), floor();
 *
 *
 * Here is the description of some local variables:
 *
 *      jk      jk+1 is the initial number of terms of ipio2[] needed
 *              in the computation. The recommended value is 2,3,4,
 *              6 for single, double, extended,and quad.
 *
 *      jz      local integer variable indicating the number of
 *              terms of ipio2[] used.
 *
 *      jx      nx - 1
 *
 *      jv      index for pointing to the suitable ipio2[] for the
 *              computation. In general, we want
 *                      ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
 *              is an integer. Thus
 *                      e0-3-24*jv >= 0 or (e0-3)/24 >= jv
 *              Hence jv = max(0,(e0-3)/24).
 *
 *      jp      jp+1 is the number of terms in PIo2[] needed, jp = jk.
 *
 *      q[]     double array with integral value, representing the
 *              24-bits chunk of the product of x and 2/pi.
 *
 *      q0      the corresponding exponent of q[0]. Note that the
 *              exponent for q[i] would be q0-24*i.
 *
 *      PIo2[]  double precision array, obtained by cutting pi/2
 *              into 24 bits chunks.
 *
 *      f[]     ipio2[] in floating point
 *
 *      iq[]    integer array by breaking up q[] in 24-bits chunk.
 *
 *      fq[]    final product of x*(2/pi) in fq[0],..,fq[jk]
 *
 *      ih      integer. If >0 it indicates q[] is >= 0.5, hence
 *              it also indicates the *sign* of the result.
 *
 */
            int __kernel_rem_pio2(double* x, double* y, int e0, int nx, int prec,
                const int32_t* ipio2)
            {
                /* Constants:
   * The hexadecimal values are the intended ones for the following
   * constants. The decimal values may be used, provided that the
   * compiler will convert from decimal to binary accurately enough
   * to produce the hexadecimal values shown.
   */
                static const int init_jk[] = { 2, 3, 4, 6 }; /* initial value for jk */

                static const double PIo2[] = {
                    1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
                    7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
                    5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
                    3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
                    1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
                    1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
                    2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
                    2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
                };

                static const double
                    zero
                    = 0.0,
                    one = 1.0,
                    two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
                    twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */

                int32_t jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih;
                double z, fw, f[20], fq[20], q[20];

                /* initialize jk*/
                jk = init_jk[prec];
                jp = jk;

                /* determine jx,jv,q0, note that 3>q0 */
                jx = nx - 1;
                jv = (e0 - 3) / 24;
                if (jv < 0)
                    jv = 0;
                q0 = e0 - 24 * (jv + 1);

                /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
                j = jv - jx;
                m = jx + jk;
                for (i = 0; i <= m; i++, j++) {
                    f[i] = (j < 0) ? zero : static_cast<double>(ipio2[j]);
                }

                /* compute q[0],q[1],...q[jk] */
                for (i = 0; i <= jk; i++) {
                    for (j = 0, fw = 0.0; j <= jx; j++)
                        fw += x[j] * f[jx + i - j];
                    q[i] = fw;
                }

                jz = jk;
            recompute:
                /* distill q[] into iq[] reversingly */
                for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--) {
                    fw = static_cast<double>(static_cast<int32_t>(twon24 * z));
                    iq[i] = static_cast<int32_t>(z - two24 * fw);
                    z = q[j - 1] + fw;
                }

                /* compute n */
                z = scalbn(z, q0); /* actual value of z */
                z -= 8.0 * floor(z * 0.125); /* trim off integer >= 8 */
                n = static_cast<int32_t>(z);
                z -= static_cast<double>(n);
                ih = 0;
                if (q0 > 0) { /* need iq[jz-1] to determine n */
                    i = (iq[jz - 1] >> (24 - q0));
                    n += i;
                    iq[jz - 1] -= i << (24 - q0);
                    ih = iq[jz - 1] >> (23 - q0);
                } else if (q0 == 0) {
                    ih = iq[jz - 1] >> 23;
                } else if (z >= 0.5) {
                    ih = 2;
                }

                if (ih > 0) { /* q > 0.5 */
                    n += 1;
                    carry = 0;
                    for (i = 0; i < jz; i++) { /* compute 1-q */
                        j = iq[i];
                        if (carry == 0) {
                            if (j != 0) {
                                carry = 1;
                                iq[i] = 0x1000000 - j;
                            }
                        } else {
                            iq[i] = 0xFFFFFF - j;
                        }
                    }
                    if (q0 > 0) { /* rare case: chance is 1 in 12 */
                        switch (q0) {
                        case 1:
                            iq[jz - 1] &= 0x7FFFFF;
                            break;
                        case 2:
                            iq[jz - 1] &= 0x3FFFFF;
                            break;
                        }
                    }
                    if (ih == 2) {
                        z = one - z;
                        if (carry != 0)
                            z -= scalbn(one, q0);
                    }
                }

                /* check if recomputation is needed */
                if (z == zero) {
                    j = 0;
                    for (i = jz - 1; i >= jk; i--)
                        j |= iq[i];
                    if (j == 0) { /* need recomputation */
                        for (k = 1; jk >= k && iq[jk - k] == 0; k++) {
                            /* k = no. of terms needed */
                        }

                        for (i = jz + 1; i <= jz + k; i++) { /* add q[jz+1] to q[jz+k] */
                            f[jx + i] = ipio2[jv + i];
                            for (j = 0, fw = 0.0; j <= jx; j++)
                                fw += x[j] * f[jx + i - j];
                            q[i] = fw;
                        }
                        jz += k;
                        goto recompute;
                    }
                }

                /* chop off zero terms */
                if (z == 0.0) {
                    jz -= 1;
                    q0 -= 24;
                    while (iq[jz] == 0) {
                        jz--;
                        q0 -= 24;
                    }
                } else { /* break z into 24-bit if necessary */
                    z = scalbn(z, -q0);
                    if (z >= two24) {
                        fw = static_cast<double>(static_cast<int32_t>(twon24 * z));
                        iq[jz] = z - two24 * fw;
                        jz += 1;
                        q0 += 24;
                        iq[jz] = fw;
                    } else {
                        iq[jz] = z;
                    }
                }

                /* convert integer "bit" chunk to floating-point value */
                fw = scalbn(one, q0);
                for (i = jz; i >= 0; i--) {
                    q[i] = fw * iq[i];
                    fw *= twon24;
                }

                /* compute PIo2[0,...,jp]*q[jz,...,0] */
                for (i = jz; i >= 0; i--) {
                    for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++)
                        fw += PIo2[k] * q[i + k];
                    fq[jz - i] = fw;
                }

                /* compress fq[] into y[] */
                switch (prec) {
                case 0:
                    fw = 0.0;
                    for (i = jz; i >= 0; i--)
                        fw += fq[i];
                    y[0] = (ih == 0) ? fw : -fw;
                    break;
                case 1:
                case 2:
                    fw = 0.0;
                    for (i = jz; i >= 0; i--)
                        fw += fq[i];
                    y[0] = (ih == 0) ? fw : -fw;
                    fw = fq[0] - fw;
                    for (i = 1; i <= jz; i++)
                        fw += fq[i];
                    y[1] = (ih == 0) ? fw : -fw;
                    break;
                case 3: /* painful */
                    for (i = jz; i > 0; i--) {
                        fw = fq[i - 1] + fq[i];
                        fq[i] += fq[i - 1] - fw;
                        fq[i - 1] = fw;
                    }
                    for (i = jz; i > 1; i--) {
                        fw = fq[i - 1] + fq[i];
                        fq[i] += fq[i - 1] - fw;
                        fq[i - 1] = fw;
                    }
                    for (fw = 0.0, i = jz; i >= 2; i--)
                        fw += fq[i];
                    if (ih == 0) {
                        y[0] = fq[0];
                        y[1] = fq[1];
                        y[2] = fw;
                    } else {
                        y[0] = -fq[0];
                        y[1] = -fq[1];
                        y[2] = -fw;
                    }
                }
                return n & 7;
            }

            /* __kernel_sin( x, y, iy)
 * kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854
 * Input x is assumed to be bounded by ~pi/4 in magnitude.
 * Input y is the tail of x.
 * Input iy indicates whether y is 0. (if iy=0, y assume to be 0).
 *
 * Algorithm
 *      1. Since sin(-x) = -sin(x), we need only to consider positive x.
 *      2. if x < 2^-27 (hx<0x3E400000 0), return x with inexact if x!=0.
 *      3. sin(x) is approximated by a polynomial of degree 13 on
 *         [0,pi/4]
 *                               3            13
 *              sin(x) ~ x + S1*x + ... + S6*x
 *         where
 *
 *      |sin(x)         2     4     6     8     10     12  |     -58
 *      |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x  +S6*x   )| <= 2
 *      |  x                                               |
 *
 *      4. sin(x+y) = sin(x) + sin'(x')*y
 *                  ~ sin(x) + (1-x*x/2)*y
 *         For better accuracy, let
 *                   3      2      2      2      2
 *              r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6))))
 *         then                   3    2
 *              sin(x) = x + (S1*x + (x *(r-y/2)+y))
 */
            V8_INLINE double __kernel_sin(double x, double y, int iy)
            {
                static const double
                    half
                    = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
                    S1 = -1.66666666666666324348e-01, /* 0xBFC55555, 0x55555549 */
                    S2 = 8.33333333332248946124e-03, /* 0x3F811111, 0x1110F8A6 */
                    S3 = -1.98412698298579493134e-04, /* 0xBF2A01A0, 0x19C161D5 */
                    S4 = 2.75573137070700676789e-06, /* 0x3EC71DE3, 0x57B1FE7D */
                    S5 = -2.50507602534068634195e-08, /* 0xBE5AE5E6, 0x8A2B9CEB */
                    S6 = 1.58969099521155010221e-10; /* 0x3DE5D93A, 0x5ACFD57C */

                double z, r, v;
                int32_t ix;
                GET_HIGH_WORD(ix, x);
                ix &= 0x7FFFFFFF; /* high word of x */
                if (ix < 0x3E400000) { /* |x| < 2**-27 */
                    if (static_cast<int>(x) == 0)
                        return x;
                } /* generate inexact */
                z = x * x;
                v = z * x;
                r = S2 + z * (S3 + z * (S4 + z * (S5 + z * S6)));
                if (iy == 0) {
                    return x + v * (S1 + z * r);
                } else {
                    return x - ((z * (half * y - v * r) - y) - v * S1);
                }
            }

            /* __kernel_tan( x, y, k )
 * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
 * Input x is assumed to be bounded by ~pi/4 in magnitude.
 * Input y is the tail of x.
 * Input k indicates whether tan (if k=1) or
 * -1/tan (if k= -1) is returned.
 *
 * Algorithm
 *      1. Since tan(-x) = -tan(x), we need only to consider positive x.
 *      2. if x < 2^-28 (hx<0x3E300000 0), return x with inexact if x!=0.
 *      3. tan(x) is approximated by a odd polynomial of degree 27 on
 *         [0,0.67434]
 *                               3             27
 *              tan(x) ~ x + T1*x + ... + T13*x
 *         where
 *
 *              |tan(x)         2     4            26   |     -59.2
 *              |----- - (1+T1*x +T2*x +.... +T13*x    )| <= 2
 *              |  x                                    |
 *
 *         Note: tan(x+y) = tan(x) + tan'(x)*y
 *                        ~ tan(x) + (1+x*x)*y
 *         Therefore, for better accuracy in computing tan(x+y), let
 *                   3      2      2       2       2
 *              r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
 *         then
 *                                  3    2
 *              tan(x+y) = x + (T1*x + (x *(r+y)+y))
 *
 *      4. For x in [0.67434,pi/4],  let y = pi/4 - x, then
 *              tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
 *                     = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
 */
            double __kernel_tan(double x, double y, int iy)
            {
                static const double xxx[] = {
                    3.33333333333334091986e-01, /* 3FD55555, 55555563 */
                    1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */
                    5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */
                    2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */
                    8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */
                    3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */
                    1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */
                    5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */
                    2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */
                    7.81794442939557092300e-05, /* 3F147E88, A03792A6 */
                    7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */
                    -1.85586374855275456654e-05, /* BEF375CB, DB605373 */
                    2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */
                    /* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */
                    /* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */
                    /* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */
                };
#define one xxx[13]
#define pio4 xxx[14]
#define pio4lo xxx[15]
#define T xxx

                double z, r, v, w, s;
                int32_t ix, hx;

                GET_HIGH_WORD(hx, x); /* high word of x */
                ix = hx & 0x7FFFFFFF; /* high word of |x| */
                if (ix < 0x3E300000) { /* x < 2**-28 */
                    if (static_cast<int>(x) == 0) { /* generate inexact */
                        uint32_t low;
                        GET_LOW_WORD(low, x);
                        if (((ix | low) | (iy + 1)) == 0) {
                            return one / fabs(x);
                        } else {
                            if (iy == 1) {
                                return x;
                            } else { /* compute -1 / (x+y) carefully */
                                double a, t;

                                z = w = x + y;
                                SET_LOW_WORD(z, 0);
                                v = y - (z - x);
                                t = a = -one / w;
                                SET_LOW_WORD(t, 0);
                                s = one + t * z;
                                return t + a * (s + t * v);
                            }
                        }
                    }
                }
                if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */
                    if (hx < 0) {
                        x = -x;
                        y = -y;
                    }
                    z = pio4 - x;
                    w = pio4lo - y;
                    x = z + w;
                    y = 0.0;
                }
                z = x * x;
                w = z * z;
                /*
   * Break x^5*(T[1]+x^2*T[2]+...) into
   * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
   * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
   */
                r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + w * T[11]))));
                v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + w * T[12])))));
                s = z * x;
                r = y + z * (s * (r + v) + y);
                r += T[0] * s;
                w = x + r;
                if (ix >= 0x3FE59428) {
                    v = iy;
                    return (1 - ((hx >> 30) & 2)) * (v - 2.0 * (x - (w * w / (w + v) - r)));
                }
                if (iy == 1) {
                    return w;
                } else {
                    /*
     * if allow error up to 2 ulp, simply return
     * -1.0 / (x+r) here
     */
                    /* compute -1.0 / (x+r) accurately */
                    double a, t;
                    z = w;
                    SET_LOW_WORD(z, 0);
                    v = r - (z - x); /* z+v = r+x */
                    t = a = -1.0 / w; /* a = -1.0/w */
                    SET_LOW_WORD(t, 0);
                    s = 1.0 + t * z;
                    return t + a * (s + t * v);
                }

#undef one
#undef pio4
#undef pio4lo
#undef T
            }

        } // namespace

        /* acos(x)
 * Method :
 *      acos(x)  = pi/2 - asin(x)
 *      acos(-x) = pi/2 + asin(x)
 * For |x|<=0.5
 *      acos(x) = pi/2 - (x + x*x^2*R(x^2))     (see asin.c)
 * For x>0.5
 *      acos(x) = pi/2 - (pi/2 - 2asin(sqrt((1-x)/2)))
 *              = 2asin(sqrt((1-x)/2))
 *              = 2s + 2s*z*R(z)        ...z=(1-x)/2, s=sqrt(z)
 *              = 2f + (2c + 2s*z*R(z))
 *     where f=hi part of s, and c = (z-f*f)/(s+f) is the correction term
 *     for f so that f+c ~ sqrt(z).
 * For x<-0.5
 *      acos(x) = pi - 2asin(sqrt((1-|x|)/2))
 *              = pi - 0.5*(s+s*z*R(z)), where z=(1-|x|)/2,s=sqrt(z)
 *
 * Special cases:
 *      if x is NaN, return x itself;
 *      if |x|>1, return NaN with invalid signal.
 *
 * Function needed: sqrt
 */
        double acos(double x)
        {
            static const double
                one
                = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
                pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
                pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
                pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
                pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
                pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
                pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
                pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
                pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
                pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
                qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
                qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
                qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
                qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */

            double z, p, q, r, w, s, c, df;
            int32_t hx, ix;
            GET_HIGH_WORD(hx, x);
            ix = hx & 0x7FFFFFFF;
            if (ix >= 0x3FF00000) { /* |x| >= 1 */
                uint32_t lx;
                GET_LOW_WORD(lx, x);
                if (((ix - 0x3FF00000) | lx) == 0) { /* |x|==1 */
                    if (hx > 0)
                        return 0.0; /* acos(1) = 0  */
                    else
                        return pi + 2.0 * pio2_lo; /* acos(-1)= pi */
                }
                return std::numeric_limits<double>::signaling_NaN(); // acos(|x|>1) is NaN
            }
            if (ix < 0x3FE00000) { /* |x| < 0.5 */
                if (ix <= 0x3C600000)
                    return pio2_hi + pio2_lo; /*if|x|<2**-57*/
                z = x * x;
                p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
                q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
                r = p / q;
                return pio2_hi - (x - (pio2_lo - x * r));
            } else if (hx < 0) { /* x < -0.5 */
                z = (one + x) * 0.5;
                p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
                q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
                s = sqrt(z);
                r = p / q;
                w = r * s - pio2_lo;
                return pi - 2.0 * (s + w);
            } else { /* x > 0.5 */
                z = (one - x) * 0.5;
                s = sqrt(z);
                df = s;
                SET_LOW_WORD(df, 0);
                c = (z - df * df) / (s + df);
                p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
                q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
                r = p / q;
                w = r * s + c;
                return 2.0 * (df + w);
            }
        }

        /* acosh(x)
 * Method :
 *      Based on
 *              acosh(x) = log [ x + sqrt(x*x-1) ]
 *      we have
 *              acosh(x) := log(x)+ln2, if x is large; else
 *              acosh(x) := log(2x-1/(sqrt(x*x-1)+x)) if x>2; else
 *              acosh(x) := log1p(t+sqrt(2.0*t+t*t)); where t=x-1.
 *
 * Special cases:
 *      acosh(x) is NaN with signal if x<1.
 *      acosh(NaN) is NaN without signal.
 */
        double acosh(double x)
        {
            static const double
                one
                = 1.0,
                ln2 = 6.93147180559945286227e-01; /* 0x3FE62E42, 0xFEFA39EF */
            double t;
            int32_t hx;
            uint32_t lx;
            EXTRACT_WORDS(hx, lx, x);
            if (hx < 0x3FF00000) { /* x < 1 */
                return std::numeric_limits<double>::signaling_NaN();
            } else if (hx >= 0x41B00000) { /* x > 2**28 */
                if (hx >= 0x7FF00000) { /* x is inf of NaN */
                    return x + x;
                } else {
                    return log(x) + ln2; /* acosh(huge)=log(2x) */
                }
            } else if (((hx - 0x3FF00000) | lx) == 0) {
                return 0.0; /* acosh(1) = 0 */
            } else if (hx > 0x40000000) { /* 2**28 > x > 2 */
                t = x * x;
                return log(2.0 * x - one / (x + sqrt(t - one)));
            } else { /* 1<x<2 */
                t = x - one;
                return log1p(t + sqrt(2.0 * t + t * t));
            }
        }

        /* asin(x)
 * Method :
 *      Since  asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
 *      we approximate asin(x) on [0,0.5] by
 *              asin(x) = x + x*x^2*R(x^2)
 *      where
 *              R(x^2) is a rational approximation of (asin(x)-x)/x^3
 *      and its remez error is bounded by
 *              |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
 *
 *      For x in [0.5,1]
 *              asin(x) = pi/2-2*asin(sqrt((1-x)/2))
 *      Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
 *      then for x>0.98
 *              asin(x) = pi/2 - 2*(s+s*z*R(z))
 *                      = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
 *      For x<=0.98, let pio4_hi = pio2_hi/2, then
 *              f = hi part of s;
 *              c = sqrt(z) - f = (z-f*f)/(s+f)         ...f+c=sqrt(z)
 *      and
 *              asin(x) = pi/2 - 2*(s+s*z*R(z))
 *                      = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
 *                      = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
 *
 * Special cases:
 *      if x is NaN, return x itself;
 *      if |x|>1, return NaN with invalid signal.
 */
        double asin(double x)
        {
            static const double
                one
                = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
                huge = 1.000e+300,
                pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
                pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
                pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
                /* coefficient for R(x^2) */
                pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
                pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
                pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
                pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
                pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
                pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
                qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
                qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
                qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
                qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */

            double t, w, p, q, c, r, s;
            int32_t hx, ix;

            t = 0;
            GET_HIGH_WORD(hx, x);
            ix = hx & 0x7FFFFFFF;
            if (ix >= 0x3FF00000) { /* |x|>= 1 */
                uint32_t lx;
                GET_LOW_WORD(lx, x);
                if (((ix - 0x3FF00000) | lx) == 0) { /* asin(1)=+-pi/2 with inexact */
                    return x * pio2_hi + x * pio2_lo;
                }
                return std::numeric_limits<double>::signaling_NaN(); // asin(|x|>1) is NaN
            } else if (ix < 0x3FE00000) { /* |x|<0.5 */
                if (ix < 0x3E400000) { /* if |x| < 2**-27 */
                    if (huge + x > one)
                        return x; /* return x with inexact if x!=0*/
                } else {
                    t = x * x;
                }
                p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
                q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
                w = p / q;
                return x + x * w;
            }
            /* 1> |x|>= 0.5 */
            w = one - fabs(x);
            t = w * 0.5;
            p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
            q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
            s = sqrt(t);
            if (ix >= 0x3FEF3333) { /* if |x| > 0.975 */
                w = p / q;
                t = pio2_hi - (2.0 * (s + s * w) - pio2_lo);
            } else {
                w = s;
                SET_LOW_WORD(w, 0);
                c = (t - w * w) / (s + w);
                r = p / q;
                p = 2.0 * s * r - (pio2_lo - 2.0 * c);
                q = pio4_hi - 2.0 * w;
                t = pio4_hi - (p - q);
            }
            if (hx > 0)
                return t;
            else
                return -t;
        }
        /* asinh(x)
 * Method :
 *      Based on
 *              asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ]
 *      we have
 *      asinh(x) := x  if  1+x*x=1,
 *               := sign(x)*(log(x)+ln2)) for large |x|, else
 *               := sign(x)*log(2|x|+1/(|x|+sqrt(x*x+1))) if|x|>2, else
 *               := sign(x)*log1p(|x| + x^2/(1 + sqrt(1+x^2)))
 */
        double asinh(double x)
        {
            static const double
                one
                = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
                ln2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
                huge = 1.00000000000000000000e+300;

            double t, w;
            int32_t hx, ix;
            GET_HIGH_WORD(hx, x);
            ix = hx & 0x7FFFFFFF;
            if (ix >= 0x7FF00000)
                return x + x; /* x is inf or NaN */
            if (ix < 0x3E300000) { /* |x|<2**-28 */
                if (huge + x > one)
                    return x; /* return x inexact except 0 */
            }
            if (ix > 0x41B00000) { /* |x| > 2**28 */
                w = log(fabs(x)) + ln2;
            } else if (ix > 0x40000000) { /* 2**28 > |x| > 2.0 */
                t = fabs(x);
                w = log(2.0 * t + one / (sqrt(x * x + one) + t));
            } else { /* 2.0 > |x| > 2**-28 */
                t = x * x;
                w = log1p(fabs(x) + t / (one + sqrt(one + t)));
            }
            if (hx > 0) {
                return w;
            } else {
                return -w;
            }
        }

        /* atan(x)
 * Method
 *   1. Reduce x to positive by atan(x) = -atan(-x).
 *   2. According to the integer k=4t+0.25 chopped, t=x, the argument
 *      is further reduced to one of the following intervals and the
 *      arctangent of t is evaluated by the corresponding formula:
 *
 *      [0,7/16]      atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
 *      [7/16,11/16]  atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) )
 *      [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) )
 *      [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) )
 *      [39/16,INF]   atan(x) = atan(INF) + atan( -1/t )
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 */
        double atan(double x)
        {
            static const double atanhi[] = {
                4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */
                7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */
                9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */
                1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */
            };

            static const double atanlo[] = {
                2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */
                3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */
                1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */
                6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */
            };

            static const double aT[] = {
                3.33333333333329318027e-01, /* 0x3FD55555, 0x5555550D */
                -1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */
                1.42857142725034663711e-01, /* 0x3FC24924, 0x920083FF */
                -1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */
                9.09088713343650656196e-02, /* 0x3FB745CD, 0xC54C206E */
                -7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */
                6.66107313738753120669e-02, /* 0x3FB10D66, 0xA0D03D51 */
                -5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */
                4.97687799461593236017e-02, /* 0x3FA97B4B, 0x24760DEB */
                -3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */
                1.62858201153657823623e-02, /* 0x3F90AD3A, 0xE322DA11 */
            };

            static const double one = 1.0, huge = 1.0e300;

            double w, s1, s2, z;
            int32_t ix, hx, id;

            GET_HIGH_WORD(hx, x);
            ix = hx & 0x7FFFFFFF;
            if (ix >= 0x44100000) { /* if |x| >= 2^66 */
                uint32_t low;
                GET_LOW_WORD(low, x);
                if (ix > 0x7FF00000 || (ix == 0x7FF00000 && (low != 0)))
                    return x + x; /* NaN */
                if (hx > 0)
                    return atanhi[3] + *const_cast<volatile double*>(&atanlo[3]);
                else
                    return -atanhi[3] - *const_cast<volatile double*>(&atanlo[3]);
            }
            if (ix < 0x3FDC0000) { /* |x| < 0.4375 */
                if (ix < 0x3E400000) { /* |x| < 2^-27 */
                    if (huge + x > one)
                        return x; /* raise inexact */
                }
                id = -1;
            } else {
                x = fabs(x);
                if (ix < 0x3FF30000) { /* |x| < 1.1875 */
                    if (ix < 0x3FE60000) { /* 7/16 <=|x|<11/16 */
                        id = 0;
                        x = (2.0 * x - one) / (2.0 + x);
                    } else { /* 11/16<=|x|< 19/16 */
                        id = 1;
                        x = (x - one) / (x + one);
                    }
                } else {
                    if (ix < 0x40038000) { /* |x| < 2.4375 */
                        id = 2;
                        x = (x - 1.5) / (one + 1.5 * x);
                    } else { /* 2.4375 <= |x| < 2^66 */
                        id = 3;
                        x = -1.0 / x;
                    }
                }
            }
            /* end of argument reduction */
            z = x * x;
            w = z * z;
            /* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */
            s1 = z * (aT[0] + w * (aT[2] + w * (aT[4] + w * (aT[6] + w * (aT[8] + w * aT[10])))));
            s2 = w * (aT[1] + w * (aT[3] + w * (aT[5] + w * (aT[7] + w * aT[9]))));
            if (id < 0) {
                return x - x * (s1 + s2);
            } else {
                z = atanhi[id] - ((x * (s1 + s2) - atanlo[id]) - x);
                return (hx < 0) ? -z : z;
            }
        }

        /* atan2(y,x)
 * Method :
 *  1. Reduce y to positive by atan2(y,x)=-atan2(-y,x).
 *  2. Reduce x to positive by (if x and y are unexceptional):
 *    ARG (x+iy) = arctan(y/x)       ... if x > 0,
 *    ARG (x+iy) = pi - arctan[y/(-x)]   ... if x < 0,
 *
 * Special cases:
 *
 *  ATAN2((anything), NaN ) is NaN;
 *  ATAN2(NAN , (anything) ) is NaN;
 *  ATAN2(+-0, +(anything but NaN)) is +-0  ;
 *  ATAN2(+-0, -(anything but NaN)) is +-pi ;
 *  ATAN2(+-(anything but 0 and NaN), 0) is +-pi/2;
 *  ATAN2(+-(anything but INF and NaN), +INF) is +-0 ;
 *  ATAN2(+-(anything but INF and NaN), -INF) is +-pi;
 *  ATAN2(+-INF,+INF ) is +-pi/4 ;
 *  ATAN2(+-INF,-INF ) is +-3pi/4;
 *  ATAN2(+-INF, (anything but,0,NaN, and INF)) is +-pi/2;
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 */
        double atan2(double y, double x)
        {
            static volatile double tiny = 1.0e-300;
            static const double
                zero
                = 0.0,
                pi_o_4 = 7.8539816339744827900E-01, /* 0x3FE921FB, 0x54442D18 */
                pi_o_2 = 1.5707963267948965580E+00, /* 0x3FF921FB, 0x54442D18 */
                pi = 3.1415926535897931160E+00; /* 0x400921FB, 0x54442D18 */
            static volatile double pi_lo = 1.2246467991473531772E-16; /* 0x3CA1A626, 0x33145C07 */

            double z;
            int32_t k, m, hx, hy, ix, iy;
            uint32_t lx, ly;

            EXTRACT_WORDS(hx, lx, x);
            ix = hx & 0x7FFFFFFF;
            EXTRACT_WORDS(hy, ly, y);
            iy = hy & 0x7FFFFFFF;
            if (((ix | ((lx | NegateWithWraparound<int32_t>(lx)) >> 31)) > 0x7FF00000) || ((iy | ((ly | NegateWithWraparound<int32_t>(ly)) >> 31)) > 0x7FF00000)) {
                return x + y; /* x or y is NaN */
            }
            if ((SubWithWraparound(hx, 0x3FF00000) | lx) == 0) {
                return atan(y); /* x=1.0 */
            }
            m = ((hy >> 31) & 1) | ((hx >> 30) & 2); /* 2*sign(x)+sign(y) */

            /* when y = 0 */
            if ((iy | ly) == 0) {
                switch (m) {
                case 0:
                case 1:
                    return y; /* atan(+-0,+anything)=+-0 */
                case 2:
                    return pi + tiny; /* atan(+0,-anything) = pi */
                case 3:
                    return -pi - tiny; /* atan(-0,-anything) =-pi */
                }
            }
            /* when x = 0 */
            if ((ix | lx) == 0)
                return (hy < 0) ? -pi_o_2 - tiny : pi_o_2 + tiny;

            /* when x is INF */
            if (ix == 0x7FF00000) {
                if (iy == 0x7FF00000) {
                    switch (m) {
                    case 0:
                        return pi_o_4 + tiny; /* atan(+INF,+INF) */
                    case 1:
                        return -pi_o_4 - tiny; /* atan(-INF,+INF) */
                    case 2:
                        return 3.0 * pi_o_4 + tiny; /*atan(+INF,-INF)*/
                    case 3:
                        return -3.0 * pi_o_4 - tiny; /*atan(-INF,-INF)*/
                    }
                } else {
                    switch (m) {
                    case 0:
                        return zero; /* atan(+...,+INF) */
                    case 1:
                        return -zero; /* atan(-...,+INF) */
                    case 2:
                        return pi + tiny; /* atan(+...,-INF) */
                    case 3:
                        return -pi - tiny; /* atan(-...,-INF) */
                    }
                }
            }
            /* when y is INF */
            if (iy == 0x7FF00000)
                return (hy < 0) ? -pi_o_2 - tiny : pi_o_2 + tiny;

            /* compute y/x */
            k = (iy - ix) >> 20;
            if (k > 60) { /* |y/x| >  2**60 */
                z = pi_o_2 + 0.5 * pi_lo;
                m &= 1;
            } else if (hx < 0 && k < -60) {
                z = 0.0; /* 0 > |y|/x > -2**-60 */
            } else {
                z = atan(fabs(y / x)); /* safe to do y/x */
            }
            switch (m) {
            case 0:
                return z; /* atan(+,+) */
            case 1:
                return -z; /* atan(-,+) */
            case 2:
                return pi - (z - pi_lo); /* atan(+,-) */
            default: /* case 3 */
                return (z - pi_lo) - pi; /* atan(-,-) */
            }
        }

        /* cos(x)
 * Return cosine function of x.
 *
 * kernel function:
 *      __kernel_sin            ... sine function on [-pi/4,pi/4]
 *      __kernel_cos            ... cosine function on [-pi/4,pi/4]
 *      __ieee754_rem_pio2      ... argument reduction routine
 *
 * Method.
 *      Let S,C and T denote the sin, cos and tan respectively on
 *      [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
 *      in [-pi/4 , +pi/4], and let n = k mod 4.
 *      We have
 *
 *          n        sin(x)      cos(x)        tan(x)
 *     ----------------------------------------------------------
 *          0          S           C             T
 *          1          C          -S            -1/T
 *          2         -S          -C             T
 *          3         -C           S            -1/T
 *     ----------------------------------------------------------
 *
 * Special cases:
 *      Let trig be any of sin, cos, or tan.
 *      trig(+-INF)  is NaN, with signals;
 *      trig(NaN)    is that NaN;
 *
 * Accuracy:
 *      TRIG(x) returns trig(x) nearly rounded
 */
        double cos(double x)
        {
            double y[2], z = 0.0;
            int32_t n, ix;

            /* High word of x. */
            GET_HIGH_WORD(ix, x);

            /* |x| ~< pi/4 */
            ix &= 0x7FFFFFFF;
            if (ix <= 0x3FE921FB) {
                return __kernel_cos(x, z);
            } else if (ix >= 0x7FF00000) {
                /* cos(Inf or NaN) is NaN */
                return x - x;
            } else {
                /* argument reduction needed */
                n = __ieee754_rem_pio2(x, y);
                switch (n & 3) {
                case 0:
                    return __kernel_cos(y[0], y[1]);
                case 1:
                    return -__kernel_sin(y[0], y[1], 1);
                case 2:
                    return -__kernel_cos(y[0], y[1]);
                default:
                    return __kernel_sin(y[0], y[1], 1);
                }
            }
        }

        /* exp(x)
 * Returns the exponential of x.
 *
 * Method
 *   1. Argument reduction:
 *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
 *      Given x, find r and integer k such that
 *
 *               x = k*ln2 + r,  |r| <= 0.5*ln2.
 *
 *      Here r will be represented as r = hi-lo for better
 *      accuracy.
 *
 *   2. Approximation of exp(r) by a special rational function on
 *      the interval [0,0.34658]:
 *      Write
 *          R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
 *      We use a special Remes algorithm on [0,0.34658] to generate
 *      a polynomial of degree 5 to approximate R. The maximum error
 *      of this polynomial approximation is bounded by 2**-59. In
 *      other words,
 *          R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
 *      (where z=r*r, and the values of P1 to P5 are listed below)
 *      and
 *          |                  5          |     -59
 *          | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
 *          |                             |
 *      The computation of exp(r) thus becomes
 *                             2*r
 *              exp(r) = 1 + -------
 *                            R - r
 *                                 r*R1(r)
 *                     = 1 + r + ----------- (for better accuracy)
 *                                2 - R1(r)
 *      where
 *                               2       4             10
 *              R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
 *
 *   3. Scale back to obtain exp(x):
 *      From step 1, we have
 *         exp(x) = 2^k * exp(r)
 *
 * Special cases:
 *      exp(INF) is INF, exp(NaN) is NaN;
 *      exp(-INF) is 0, and
 *      for finite argument, only exp(0)=1 is exact.
 *
 * Accuracy:
 *      according to an error analysis, the error is always less than
 *      1 ulp (unit in the last place).
 *
 * Misc. info.
 *      For IEEE double
 *          if x >  7.09782712893383973096e+02 then exp(x) overflow
 *          if x < -7.45133219101941108420e+02 then exp(x) underflow
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 */
        double exp(double x)
        {
            static const double
                one
                = 1.0,
                halF[2] = { 0.5, -0.5 },
                o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
                u_threshold = -7.45133219101941108420e+02, /* 0xC0874910, 0xD52D3051 */
                ln2HI[2] = { 6.93147180369123816490e-01, /* 0x3FE62E42, 0xFEE00000 */
                    -6.93147180369123816490e-01 }, /* 0xBFE62E42, 0xFEE00000 */
                ln2LO[2] = { 1.90821492927058770002e-10, /* 0x3DEA39EF, 0x35793C76 */
                    -1.90821492927058770002e-10 }, /* 0xBDEA39EF, 0x35793C76 */
                invln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE */
                P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
                P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
                P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
                P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
                P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
                E = 2.718281828459045; /* 0x4005BF0A, 0x8B145769 */

            static volatile double
                huge
                = 1.0e+300,
                twom1000 = 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
                two1023 = 8.988465674311579539e307; /* 0x1p1023 */

            double y, hi = 0.0, lo = 0.0, c, t, twopk;
            int32_t k = 0, xsb;
            uint32_t hx;

            GET_HIGH_WORD(hx, x);
            xsb = (hx >> 31) & 1; /* sign bit of x */
            hx &= 0x7FFFFFFF; /* high word of |x| */

            /* filter out non-finite argument */
            if (hx >= 0x40862E42) { /* if |x|>=709.78... */
                if (hx >= 0x7FF00000) {
                    uint32_t lx;
                    GET_LOW_WORD(lx, x);
                    if (((hx & 0xFFFFF) | lx) != 0)
                        return x + x; /* NaN */
                    else
                        return (xsb == 0) ? x : 0.0; /* exp(+-inf)={inf,0} */
                }
                if (x > o_threshold)
                    return huge * huge; /* overflow */
                if (x < u_threshold)
                    return twom1000 * twom1000; /* underflow */
            }

            /* argument reduction */
            if (hx > 0x3FD62E42) { /* if  |x| > 0.5 ln2 */
                if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
                    /* TODO(rtoy): We special case exp(1) here to return the correct
       * value of E, as the computation below would get the last bit
       * wrong. We should probably fix the algorithm instead.
       */
                    if (x == 1.0)
                        return E;
                    hi = x - ln2HI[xsb];
                    lo = ln2LO[xsb];
                    k = 1 - xsb - xsb;
                } else {
                    k = static_cast<int>(invln2 * x + halF[xsb]);
                    t = k;
                    hi = x - t * ln2HI[0]; /* t*ln2HI is exact here */
                    lo = t * ln2LO[0];
                }
                STRICT_ASSIGN(double, x, hi - lo);
            } else if (hx < 0x3E300000) { /* when |x|<2**-28 */
                if (huge + x > one)
                    return one + x; /* trigger inexact */
            } else {
                k = 0;
            }

            /* x is now in primary range */
            t = x * x;
            if (k >= -1021) {
                INSERT_WORDS(
                    twopk,
                    0x3FF00000 + static_cast<int32_t>(static_cast<uint32_t>(k) << 20), 0);
            } else {
                INSERT_WORDS(twopk, 0x3FF00000 + (static_cast<uint32_t>(k + 1000) << 20),
                    0);
            }
            c = x - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
            if (k == 0) {
                return one - ((x * c) / (c - 2.0) - x);
            } else {
                y = one - ((lo - (x * c) / (2.0 - c)) - hi);
            }
            if (k >= -1021) {
                if (k == 1024)
                    return y * 2.0 * two1023;
                return y * twopk;
            } else {
                return y * twopk * twom1000;
            }
        }

        /*
 * Method :
 *    1.Reduced x to positive by atanh(-x) = -atanh(x)
 *    2.For x>=0.5
 *              1              2x                          x
 *  atanh(x) = --- * log(1 + -------) = 0.5 * log1p(2 * --------)
 *              2             1 - x                      1 - x
 *
 *   For x<0.5
 *  atanh(x) = 0.5*log1p(2x+2x*x/(1-x))
 *
 * Special cases:
 *  atanh(x) is NaN if |x| > 1 with signal;
 *  atanh(NaN) is that NaN with no signal;
 *  atanh(+-1) is +-INF with signal.
 *
 */
        double atanh(double x)
        {
            static const double one = 1.0, huge = 1e300;
            static const double zero = 0.0;

            double t;
            int32_t hx, ix;
            uint32_t lx;
            EXTRACT_WORDS(hx, lx, x);
            ix = hx & 0x7FFFFFFF;
            if ((ix | ((lx | NegateWithWraparound<int32_t>(lx)) >> 31)) > 0x3FF00000) {
                /* |x|>1 */
                return std::numeric_limits<double>::signaling_NaN();
            }
            if (ix == 0x3FF00000) {
                return x > 0 ? std::numeric_limits<double>::infinity()
                             : -std::numeric_limits<double>::infinity();
            }
            if (ix < 0x3E300000 && (huge + x) > zero)
                return x; /* x<2**-28 */
            SET_HIGH_WORD(x, ix);
            if (ix < 0x3FE00000) { /* x < 0.5 */
                t = x + x;
                t = 0.5 * log1p(t + t * x / (one - x));
            } else {
                t = 0.5 * log1p((x + x) / (one - x));
            }
            if (hx >= 0)
                return t;
            else
                return -t;
        }

        /* log(x)
 * Return the logrithm of x
 *
 * Method :
 *   1. Argument Reduction: find k and f such that
 *     x = 2^k * (1+f),
 *     where  sqrt(2)/2 < 1+f < sqrt(2) .
 *
 *   2. Approximation of log(1+f).
 *  Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
 *     = 2s + 2/3 s**3 + 2/5 s**5 + .....,
 *         = 2s + s*R
 *      We use a special Reme algorithm on [0,0.1716] to generate
 *  a polynomial of degree 14 to approximate R The maximum error
 *  of this polynomial approximation is bounded by 2**-58.45. In
 *  other words,
 *            2      4      6      8      10      12      14
 *      R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
 *    (the values of Lg1 to Lg7 are listed in the program)
 *  and
 *      |      2          14          |     -58.45
 *      | Lg1*s +...+Lg7*s    -  R(z) | <= 2
 *      |                             |
 *  Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
 *  In order to guarantee error in log below 1ulp, we compute log
 *  by
 *    log(1+f) = f - s*(f - R)  (if f is not too large)
 *    log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
 *
 *  3. Finally,  log(x) = k*ln2 + log(1+f).
 *          = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
 *     Here ln2 is split into two floating point number:
 *      ln2_hi + ln2_lo,
 *     where n*ln2_hi is always exact for |n| < 2000.
 *
 * Special cases:
 *  log(x) is NaN with signal if x < 0 (including -INF) ;
 *  log(+INF) is +INF; log(0) is -INF with signal;
 *  log(NaN) is that NaN with no signal.
 *
 * Accuracy:
 *  according to an error analysis, the error is always less than
 *  1 ulp (unit in the last place).
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 */
        double log(double x)
        {
            static const double /* -- */
                ln2_hi
                = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
                ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
                two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
                Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
                Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
                Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
                Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
                Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
                Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
                Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */

            static const double zero = 0.0;

            double hfsq, f, s, z, R, w, t1, t2, dk;
            int32_t k, hx, i, j;
            uint32_t lx;

            EXTRACT_WORDS(hx, lx, x);

            k = 0;
            if (hx < 0x00100000) { /* x < 2**-1022  */
                if (((hx & 0x7FFFFFFF) | lx) == 0) {
                    return -std::numeric_limits<double>::infinity(); /* log(+-0)=-inf */
                }
                if (hx < 0) {
                    return std::numeric_limits<double>::signaling_NaN(); /* log(-#) = NaN */
                }
                k -= 54;
                x *= two54; /* subnormal number, scale up x */
                GET_HIGH_WORD(hx, x);
            }
            if (hx >= 0x7FF00000)
                return x + x;
            k += (hx >> 20) - 1023;
            hx &= 0x000FFFFF;
            i = (hx + 0x95F64) & 0x100000;
            SET_HIGH_WORD(x, hx | (i ^ 0x3FF00000)); /* normalize x or x/2 */
            k += (i >> 20);
            f = x - 1.0;
            if ((0x000FFFFF & (2 + hx)) < 3) { /* -2**-20 <= f < 2**-20 */
                if (f == zero) {
                    if (k == 0) {
                        return zero;
                    } else {
                        dk = static_cast<double>(k);
                        return dk * ln2_hi + dk * ln2_lo;
                    }
                }
                R = f * f * (0.5 - 0.33333333333333333 * f);
                if (k == 0) {
                    return f - R;
                } else {
                    dk = static_cast<double>(k);
                    return dk * ln2_hi - ((R - dk * ln2_lo) - f);
                }
            }
            s = f / (2.0 + f);
            dk = static_cast<double>(k);
            z = s * s;
            i = hx - 0x6147A;
            w = z * z;
            j = 0x6B851 - hx;
            t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
            t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
            i |= j;
            R = t2 + t1;
            if (i > 0) {
                hfsq = 0.5 * f * f;
                if (k == 0)
                    return f - (hfsq - s * (hfsq + R));
                else
                    return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) - f);
            } else {
                if (k == 0)
                    return f - s * (f - R);
                else
                    return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f);
            }
        }

        /* double log1p(double x)
 *
 * Method :
 *   1. Argument Reduction: find k and f such that
 *      1+x = 2^k * (1+f),
 *     where  sqrt(2)/2 < 1+f < sqrt(2) .
 *
 *      Note. If k=0, then f=x is exact. However, if k!=0, then f
 *  may not be representable exactly. In that case, a correction
 *  term is need. Let u=1+x rounded. Let c = (1+x)-u, then
 *  log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
 *  and add back the correction term c/u.
 *  (Note: when x > 2**53, one can simply return log(x))
 *
 *   2. Approximation of log1p(f).
 *  Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
 *     = 2s + 2/3 s**3 + 2/5 s**5 + .....,
 *         = 2s + s*R
 *      We use a special Reme algorithm on [0,0.1716] to generate
 *  a polynomial of degree 14 to approximate R The maximum error
 *  of this polynomial approximation is bounded by 2**-58.45. In
 *  other words,
 *            2      4      6      8      10      12      14
 *      R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s  +Lp6*s  +Lp7*s
 *    (the values of Lp1 to Lp7 are listed in the program)
 *  and
 *      |      2          14          |     -58.45
 *      | Lp1*s +...+Lp7*s    -  R(z) | <= 2
 *      |                             |
 *  Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
 *  In order to guarantee error in log below 1ulp, we compute log
 *  by
 *    log1p(f) = f - (hfsq - s*(hfsq+R)).
 *
 *  3. Finally, log1p(x) = k*ln2 + log1p(f).
 *           = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
 *     Here ln2 is split into two floating point number:
 *      ln2_hi + ln2_lo,
 *     where n*ln2_hi is always exact for |n| < 2000.
 *
 * Special cases:
 *  log1p(x) is NaN with signal if x < -1 (including -INF) ;
 *  log1p(+INF) is +INF; log1p(-1) is -INF with signal;
 *  log1p(NaN) is that NaN with no signal.
 *
 * Accuracy:
 *  according to an error analysis, the error is always less than
 *  1 ulp (unit in the last place).
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 *
 * Note: Assuming log() return accurate answer, the following
 *   algorithm can be used to compute log1p(x) to within a few ULP:
 *
 *    u = 1+x;
 *    if(u==1.0) return x ; else
 *         return log(u)*(x/(u-1.0));
 *
 *   See HP-15C Advanced Functions Handbook, p.193.
 */
        double log1p(double x)
        {
            static const double /* -- */
                ln2_hi
                = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
                ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
                two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
                Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
                Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
                Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
                Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
                Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
                Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
                Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */

            static const double zero = 0.0;

            double hfsq, f, c, s, z, R, u;
            int32_t k, hx, hu, ax;

            GET_HIGH_WORD(hx, x);
            ax = hx & 0x7FFFFFFF;

            k = 1;
            if (hx < 0x3FDA827A) { /* 1+x < sqrt(2)+ */
                if (ax >= 0x3FF00000) { /* x <= -1.0 */
                    if (x == -1.0)
                        return -std::numeric_limits<double>::infinity(); /* log1p(-1)=+inf */
                    else
                        return std::numeric_limits<double>::signaling_NaN(); // log1p(x<-1)=NaN
                }
                if (ax < 0x3E200000) { /* |x| < 2**-29 */
                    if (two54 + x > zero /* raise inexact */
                        && ax < 0x3C900000) /* |x| < 2**-54 */
                        return x;
                    else
                        return x - x * x * 0.5;
                }
                if (hx > 0 || hx <= static_cast<int32_t>(0xBFD2BEC4)) {
                    k = 0;
                    f = x;
                    hu = 1;
                } /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
            }
            if (hx >= 0x7FF00000)
                return x + x;
            if (k != 0) {
                if (hx < 0x43400000) {
                    STRICT_ASSIGN(double, u, 1.0 + x);
                    GET_HIGH_WORD(hu, u);
                    k = (hu >> 20) - 1023;
                    c = (k > 0) ? 1.0 - (u - x) : x - (u - 1.0); /* correction term */
                    c /= u;
                } else {
                    u = x;
                    GET_HIGH_WORD(hu, u);
                    k = (hu >> 20) - 1023;
                    c = 0;
                }
                hu &= 0x000FFFFF;
                /*
     * The approximation to sqrt(2) used in thresholds is not
     * critical.  However, the ones used above must give less
     * strict bounds than the one here so that the k==0 case is
     * never reached from here, since here we have committed to
     * using the correction term but don't use it if k==0.
     */
                if (hu < 0x6A09E) { /* u ~< sqrt(2) */
                    SET_HIGH_WORD(u, hu | 0x3FF00000); /* normalize u */
                } else {
                    k += 1;
                    SET_HIGH_WORD(u, hu | 0x3FE00000); /* normalize u/2 */
                    hu = (0x00100000 - hu) >> 2;
                }
                f = u - 1.0;
            }
            hfsq = 0.5 * f * f;
            if (hu == 0) { /* |f| < 2**-20 */
                if (f == zero) {
                    if (k == 0) {
                        return zero;
                    } else {
                        c += k * ln2_lo;
                        return k * ln2_hi + c;
                    }
                }
                R = hfsq * (1.0 - 0.66666666666666666 * f);
                if (k == 0)
                    return f - R;
                else
                    return k * ln2_hi - ((R - (k * ln2_lo + c)) - f);
            }
            s = f / (2.0 + f);
            z = s * s;
            R = z * (Lp1 + z * (Lp2 + z * (Lp3 + z * (Lp4 + z * (Lp5 + z * (Lp6 + z * Lp7))))));
            if (k == 0)
                return f - (hfsq - s * (hfsq + R));
            else
                return k * ln2_hi - ((hfsq - (s * (hfsq + R) + (k * ln2_lo + c))) - f);
        }

        /*
 * k_log1p(f):
 * Return log(1+f) - f for 1+f in ~[sqrt(2)/2, sqrt(2)].
 *
 * The following describes the overall strategy for computing
 * logarithms in base e.  The argument reduction and adding the final
 * term of the polynomial are done by the caller for increased accuracy
 * when different bases are used.
 *
 * Method :
 *   1. Argument Reduction: find k and f such that
 *         x = 2^k * (1+f),
 *         where  sqrt(2)/2 < 1+f < sqrt(2) .
 *
 *   2. Approximation of log(1+f).
 *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
 *            = 2s + 2/3 s**3 + 2/5 s**5 + .....,
 *            = 2s + s*R
 *      We use a special Reme algorithm on [0,0.1716] to generate
 *      a polynomial of degree 14 to approximate R The maximum error
 *      of this polynomial approximation is bounded by 2**-58.45. In
 *      other words,
 *          2      4      6      8      10      12      14
 *          R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
 *      (the values of Lg1 to Lg7 are listed in the program)
 *      and
 *          |      2          14          |     -58.45
 *          | Lg1*s +...+Lg7*s    -  R(z) | <= 2
 *          |                             |
 *      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
 *      In order to guarantee error in log below 1ulp, we compute log
 *      by
 *          log(1+f) = f - s*(f - R)            (if f is not too large)
 *          log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
 *
 *   3. Finally,  log(x) = k*ln2 + log(1+f).
 *          = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
 *      Here ln2 is split into two floating point number:
 *          ln2_hi + ln2_lo,
 *      where n*ln2_hi is always exact for |n| < 2000.
 *
 * Special cases:
 *      log(x) is NaN with signal if x < 0 (including -INF) ;
 *      log(+INF) is +INF; log(0) is -INF with signal;
 *      log(NaN) is that NaN with no signal.
 *
 * Accuracy:
 *      according to an error analysis, the error is always less than
 *      1 ulp (unit in the last place).
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 */

        static const double Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
            Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
            Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
            Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
            Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
            Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
            Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */

        /*
 * We always inline k_log1p(), since doing so produces a
 * substantial performance improvement (~40% on amd64).
 */
        static inline double k_log1p(double f)
        {
            double hfsq, s, z, R, w, t1, t2;

            s = f / (2.0 + f);
            z = s * s;
            w = z * z;
            t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
            t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
            R = t2 + t1;
            hfsq = 0.5 * f * f;
            return s * (hfsq + R);
        }

        /*
 * Return the base 2 logarithm of x.  See e_log.c and k_log.h for most
 * comments.
 *
 * This reduces x to {k, 1+f} exactly as in e_log.c, then calls the kernel,
 * then does the combining and scaling steps
 *    log2(x) = (f - 0.5*f*f + k_log1p(f)) / ln2 + k
 * in not-quite-routine extra precision.
 */
        double log2(double x)
        {
            static const double
                two54
                = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
                ivln2hi = 1.44269504072144627571e+00, /* 0x3FF71547, 0x65200000 */
                ivln2lo = 1.67517131648865118353e-10; /* 0x3DE705FC, 0x2EEFA200 */

            double f, hfsq, hi, lo, r, val_hi, val_lo, w, y;
            int32_t i, k, hx;
            uint32_t lx;

            EXTRACT_WORDS(hx, lx, x);

            k = 0;
            if (hx < 0x00100000) { /* x < 2**-1022  */
                if (((hx & 0x7FFFFFFF) | lx) == 0) {
                    return -std::numeric_limits<double>::infinity(); /* log(+-0)=-inf */
                }
                if (hx < 0) {
                    return std::numeric_limits<double>::signaling_NaN(); /* log(-#) = NaN */
                }
                k -= 54;
                x *= two54; /* subnormal number, scale up x */
                GET_HIGH_WORD(hx, x);
            }
            if (hx >= 0x7FF00000)
                return x + x;
            if (hx == 0x3FF00000 && lx == 0)
                return 0.0; /* log(1) = +0 */
            k += (hx >> 20) - 1023;
            hx &= 0x000FFFFF;
            i = (hx + 0x95F64) & 0x100000;
            SET_HIGH_WORD(x, hx | (i ^ 0x3FF00000)); /* normalize x or x/2 */
            k += (i >> 20);
            y = static_cast<double>(k);
            f = x - 1.0;
            hfsq = 0.5 * f * f;
            r = k_log1p(f);

            /*
   * f-hfsq must (for args near 1) be evaluated in extra precision
   * to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2).
   * This is fairly efficient since f-hfsq only depends on f, so can
   * be evaluated in parallel with R.  Not combining hfsq with R also
   * keeps R small (though not as small as a true `lo' term would be),
   * so that extra precision is not needed for terms involving R.
   *
   * Compiler bugs involving extra precision used to break Dekker's
   * theorem for spitting f-hfsq as hi+lo, unless double_t was used
   * or the multi-precision calculations were avoided when double_t
   * has extra precision.  These problems are now automatically
   * avoided as a side effect of the optimization of combining the
   * Dekker splitting step with the clear-low-bits step.
   *
   * y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra
   * precision to avoid a very large cancellation when x is very near
   * these values.  Unlike the above cancellations, this problem is
   * specific to base 2.  It is strange that adding +-1 is so much
   * harder than adding +-ln2 or +-log10_2.
   *
   * This uses Dekker's theorem to normalize y+val_hi, so the
   * compiler bugs are back in some configurations, sigh.  And I
   * don't want to used double_t to avoid them, since that gives a
   * pessimization and the support for avoiding the pessimization
   * is not yet available.
   *
   * The multi-precision calculations for the multiplications are
   * routine.
   */
            hi = f - hfsq;
            SET_LOW_WORD(hi, 0);
            lo = (f - hi) - hfsq + r;
            val_hi = hi * ivln2hi;
            val_lo = (lo + hi) * ivln2lo + lo * ivln2hi;

            /* spadd(val_hi, val_lo, y), except for not using double_t: */
            w = y + val_hi;
            val_lo += (y - w) + val_hi;
            val_hi = w;

            return val_lo + val_hi;
        }

        /*
 * Return the base 10 logarithm of x
 *
 * Method :
 *      Let log10_2hi = leading 40 bits of log10(2) and
 *          log10_2lo = log10(2) - log10_2hi,
 *          ivln10   = 1/log(10) rounded.
 *      Then
 *              n = ilogb(x),
 *              if(n<0)  n = n+1;
 *              x = scalbn(x,-n);
 *              log10(x) := n*log10_2hi + (n*log10_2lo + ivln10*log(x))
 *
 *  Note 1:
 *     To guarantee log10(10**n)=n, where 10**n is normal, the rounding
 *     mode must set to Round-to-Nearest.
 *  Note 2:
 *      [1/log(10)] rounded to 53 bits has error .198 ulps;
 *      log10 is monotonic at all binary break points.
 *
 *  Special cases:
 *      log10(x) is NaN if x < 0;
 *      log10(+INF) is +INF; log10(0) is -INF;
 *      log10(NaN) is that NaN;
 *      log10(10**N) = N  for N=0,1,...,22.
 */
        double log10(double x)
        {
            static const double
                two54
                = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
                ivln10 = 4.34294481903251816668e-01,
                log10_2hi = 3.01029995663611771306e-01, /* 0x3FD34413, 0x509F6000 */
                log10_2lo = 3.69423907715893078616e-13; /* 0x3D59FEF3, 0x11F12B36 */

            double y;
            int32_t i, k, hx;
            uint32_t lx;

            EXTRACT_WORDS(hx, lx, x);

            k = 0;
            if (hx < 0x00100000) { /* x < 2**-1022  */
                if (((hx & 0x7FFFFFFF) | lx) == 0) {
                    return -std::numeric_limits<double>::infinity(); /* log(+-0)=-inf */
                }
                if (hx < 0) {
                    return std::numeric_limits<double>::quiet_NaN(); /* log(-#) = NaN */
                }
                k -= 54;
                x *= two54; /* subnormal number, scale up x */
                GET_HIGH_WORD(hx, x);
                GET_LOW_WORD(lx, x);
            }
            if (hx >= 0x7FF00000)
                return x + x;
            if (hx == 0x3FF00000 && lx == 0)
                return 0.0; /* log(1) = +0 */
            k += (hx >> 20) - 1023;

            i = (k & 0x80000000) >> 31;
            hx = (hx & 0x000FFFFF) | ((0x3FF - i) << 20);
            y = k + i;
            SET_HIGH_WORD(x, hx);
            SET_LOW_WORD(x, lx);

            double z = y * log10_2lo + ivln10 * log(x);
            return z + y * log10_2hi;
        }

        /* expm1(x)
 * Returns exp(x)-1, the exponential of x minus 1.
 *
 * Method
 *   1. Argument reduction:
 *  Given x, find r and integer k such that
 *
 *               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658
 *
 *      Here a correction term c will be computed to compensate
 *  the error in r when rounded to a floating-point number.
 *
 *   2. Approximating expm1(r) by a special rational function on
 *  the interval [0,0.34658]:
 *  Since
 *      r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
 *  we define R1(r*r) by
 *      r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
 *  That is,
 *      R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
 *         = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
 *         = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
 *      We use a special Reme algorithm on [0,0.347] to generate
 *   a polynomial of degree 5 in r*r to approximate R1. The
 *  maximum error of this polynomial approximation is bounded
 *  by 2**-61. In other words,
 *      R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
 *  where   Q1  =  -1.6666666666666567384E-2,
 *     Q2  =   3.9682539681370365873E-4,
 *     Q3  =  -9.9206344733435987357E-6,
 *     Q4  =   2.5051361420808517002E-7,
 *     Q5  =  -6.2843505682382617102E-9;
 *    z   =  r*r,
 *  with error bounded by
 *      |                  5           |     -61
 *      | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2
 *      |                              |
 *
 *  expm1(r) = exp(r)-1 is then computed by the following
 *   specific way which minimize the accumulation rounding error:
 *             2     3
 *            r     r    [ 3 - (R1 + R1*r/2)  ]
 *        expm1(r) = r + --- + --- * [--------------------]
 *                  2     2    [ 6 - r*(3 - R1*r/2) ]
 *
 *  To compensate the error in the argument reduction, we use
 *    expm1(r+c) = expm1(r) + c + expm1(r)*c
 *         ~ expm1(r) + c + r*c
 *  Thus c+r*c will be added in as the correction terms for
 *  expm1(r+c). Now rearrange the term to avoid optimization
 *   screw up:
 *            (      2                                    2 )
 *            ({  ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )
 *   expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
 *                  ({  ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )
 *                      (                                             )
 *
 *       = r - E
 *   3. Scale back to obtain expm1(x):
 *  From step 1, we have
 *     expm1(x) = either 2^k*[expm1(r)+1] - 1
 *        = or     2^k*[expm1(r) + (1-2^-k)]
 *   4. Implementation notes:
 *  (A). To save one multiplication, we scale the coefficient Qi
 *       to Qi*2^i, and replace z by (x^2)/2.
 *  (B). To achieve maximum accuracy, we compute expm1(x) by
 *    (i)   if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
 *    (ii)  if k=0, return r-E
 *    (iii) if k=-1, return 0.5*(r-E)-0.5
 *        (iv)  if k=1 if r < -0.25, return 2*((r+0.5)- E)
 *                  else       return  1.0+2.0*(r-E);
 *    (v)   if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
 *    (vi)  if k <= 20, return 2^k((1-2^-k)-(E-r)), else
 *    (vii) return 2^k(1-((E+2^-k)-r))
 *
 * Special cases:
 *  expm1(INF) is INF, expm1(NaN) is NaN;
 *  expm1(-INF) is -1, and
 *  for finite argument, only expm1(0)=0 is exact.
 *
 * Accuracy:
 *  according to an error analysis, the error is always less than
 *  1 ulp (unit in the last place).
 *
 * Misc. info.
 *  For IEEE double
 *      if x >  7.09782712893383973096e+02 then expm1(x) overflow
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 */
        double expm1(double x)
        {
            static const double
                one
                = 1.0,
                tiny = 1.0e-300,
                o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
                ln2_hi = 6.93147180369123816490e-01, /* 0x3FE62E42, 0xFEE00000 */
                ln2_lo = 1.90821492927058770002e-10, /* 0x3DEA39EF, 0x35793C76 */
                invln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE */
                /* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs =
         x*x/2: */
                Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
                Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
                Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
                Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
                Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */

            static volatile double huge = 1.0e+300;

            double y, hi, lo, c, t, e, hxs, hfx, r1, twopk;
            int32_t k, xsb;
            uint32_t hx;

            GET_HIGH_WORD(hx, x);
            xsb = hx & 0x80000000; /* sign bit of x */
            hx &= 0x7FFFFFFF; /* high word of |x| */

            /* filter out huge and non-finite argument */
            if (hx >= 0x4043687A) { /* if |x|>=56*ln2 */
                if (hx >= 0x40862E42) { /* if |x|>=709.78... */
                    if (hx >= 0x7FF00000) {
                        uint32_t low;
                        GET_LOW_WORD(low, x);
                        if (((hx & 0xFFFFF) | low) != 0)
                            return x + x; /* NaN */
                        else
                            return (xsb == 0) ? x : -1.0; /* exp(+-inf)={inf,-1} */
                    }
                    if (x > o_threshold)
                        return huge * huge; /* overflow */
                }
                if (xsb != 0) { /* x < -56*ln2, return -1.0 with inexact */
                    if (x + tiny < 0.0) /* raise inexact */
                        return tiny - one; /* return -1 */
                }
            }

            /* argument reduction */
            if (hx > 0x3FD62E42) { /* if  |x| > 0.5 ln2 */
                if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
                    if (xsb == 0) {
                        hi = x - ln2_hi;
                        lo = ln2_lo;
                        k = 1;
                    } else {
                        hi = x + ln2_hi;
                        lo = -ln2_lo;
                        k = -1;
                    }
                } else {
                    k = invln2 * x + ((xsb == 0) ? 0.5 : -0.5);
                    t = k;
                    hi = x - t * ln2_hi; /* t*ln2_hi is exact here */
                    lo = t * ln2_lo;
                }
                STRICT_ASSIGN(double, x, hi - lo);
                c = (hi - x) - lo;
            } else if (hx < 0x3C900000) { /* when |x|<2**-54, return x */
                t = huge + x; /* return x with inexact flags when x!=0 */
                return x - (t - (huge + x));
            } else {
                k = 0;
            }

            /* x is now in primary range */
            hfx = 0.5 * x;
            hxs = x * hfx;
            r1 = one + hxs * (Q1 + hxs * (Q2 + hxs * (Q3 + hxs * (Q4 + hxs * Q5))));
            t = 3.0 - r1 * hfx;
            e = hxs * ((r1 - t) / (6.0 - x * t));
            if (k == 0) {
                return x - (x * e - hxs); /* c is 0 */
            } else {
                INSERT_WORDS(
                    twopk,
                    0x3FF00000 + static_cast<int32_t>(static_cast<uint32_t>(k) << 20),
                    0); /* 2^k */
                e = (x * (e - c) - c);
                e -= hxs;
                if (k == -1)
                    return 0.5 * (x - e) - 0.5;
                if (k == 1) {
                    if (x < -0.25)
                        return -2.0 * (e - (x + 0.5));
                    else
                        return one + 2.0 * (x - e);
                }
                if (k <= -2 || k > 56) { /* suffice to return exp(x)-1 */
                    y = one - (e - x);
                    // TODO(mvstanton): is this replacement for the hex float
                    // sufficient?
                    // if (k == 1024) y = y*2.0*0x1p1023;
                    if (k == 1024)
                        y = y * 2.0 * 8.98846567431158e+307;
                    else
                        y = y * twopk;
                    return y - one;
                }
                t = one;
                if (k < 20) {
                    SET_HIGH_WORD(t, 0x3FF00000 - (0x200000 >> k)); /* t=1-2^-k */
                    y = t - (e - x);
                    y = y * twopk;
                } else {
                    SET_HIGH_WORD(t, ((0x3FF - k) << 20)); /* 2^-k */
                    y = x - (e + t);
                    y += one;
                    y = y * twopk;
                }
            }
            return y;
        }

        double cbrt(double x)
        {
            static const uint32_t
                B1
                = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */
                B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */

            /* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */
            static const double P0 = 1.87595182427177009643, /* 0x3FFE03E6, 0x0F61E692 */
                P1 = -1.88497979543377169875, /* 0xBFFE28E0, 0x92F02420 */
                P2 = 1.621429720105354466140, /* 0x3FF9F160, 0x4A49D6C2 */
                P3 = -0.758397934778766047437, /* 0xBFE844CB, 0xBEE751D9 */
                P4 = 0.145996192886612446982; /* 0x3FC2B000, 0xD4E4EDD7 */

            int32_t hx;
            union {
                double value;
                uint64_t bits;
            } u;
            double r, s, t = 0.0, w;
            uint32_t sign;
            uint32_t high, low;

            EXTRACT_WORDS(hx, low, x);
            sign = hx & 0x80000000; /* sign= sign(x) */
            hx ^= sign;
            if (hx >= 0x7FF00000)
                return (x + x); /* cbrt(NaN,INF) is itself */

            /*
   * Rough cbrt to 5 bits:
   *    cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3)
   * where e is integral and >= 0, m is real and in [0, 1), and "/" and
   * "%" are integer division and modulus with rounding towards minus
   * infinity.  The RHS is always >= the LHS and has a maximum relative
   * error of about 1 in 16.  Adding a bias of -0.03306235651 to the
   * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE
   * floating point representation, for finite positive normal values,
   * ordinary integer division of the value in bits magically gives
   * almost exactly the RHS of the above provided we first subtract the
   * exponent bias (1023 for doubles) and later add it back.  We do the
   * subtraction virtually to keep e >= 0 so that ordinary integer
   * division rounds towards minus infinity; this is also efficient.
   */
            if (hx < 0x00100000) { /* zero or subnormal? */
                if ((hx | low) == 0)
                    return (x); /* cbrt(0) is itself */
                SET_HIGH_WORD(t, 0x43500000); /* set t= 2**54 */
                t *= x;
                GET_HIGH_WORD(high, t);
                INSERT_WORDS(t, sign | ((high & 0x7FFFFFFF) / 3 + B2), 0);
            } else {
                INSERT_WORDS(t, sign | (hx / 3 + B1), 0);
            }

            /*
   * New cbrt to 23 bits:
   *    cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x)
   * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r)
   * to within 2**-23.5 when |r - 1| < 1/10.  The rough approximation
   * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this
   * gives us bounds for r = t**3/x.
   *
   * Try to optimize for parallel evaluation as in k_tanf.c.
   */
            r = (t * t) * (t / x);
            t = t * ((P0 + r * (P1 + r * P2)) + ((r * r) * r) * (P3 + r * P4));

            /*
   * Round t away from zero to 23 bits (sloppily except for ensuring that
   * the result is larger in magnitude than cbrt(x) but not much more than
   * 2 23-bit ulps larger).  With rounding towards zero, the error bound
   * would be ~5/6 instead of ~4/6.  With a maximum error of 2 23-bit ulps
   * in the rounded t, the infinite-precision error in the Newton
   * approximation barely affects third digit in the final error
   * 0.667; the error in the rounded t can be up to about 3 23-bit ulps
   * before the final error is larger than 0.667 ulps.
   */
            u.value = t;
            u.bits = (u.bits + 0x80000000) & 0xFFFFFFFFC0000000ULL;
            t = u.value;

            /* one step Newton iteration to 53 bits with error < 0.667 ulps */
            s = t * t; /* t*t is exact */
            r = x / s; /* error <= 0.5 ulps; |r| < |t| */
            w = t + t; /* t+t is exact */
            r = (r - t) / (w + r); /* r-t is exact; w+r ~= 3*t */
            t = t + t * r; /* error <= 0.5 + 0.5/3 + epsilon */

            return (t);
        }

        /* sin(x)
 * Return sine function of x.
 *
 * kernel function:
 *      __kernel_sin            ... sine function on [-pi/4,pi/4]
 *      __kernel_cos            ... cose function on [-pi/4,pi/4]
 *      __ieee754_rem_pio2      ... argument reduction routine
 *
 * Method.
 *      Let S,C and T denote the sin, cos and tan respectively on
 *      [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
 *      in [-pi/4 , +pi/4], and let n = k mod 4.
 *      We have
 *
 *          n        sin(x)      cos(x)        tan(x)
 *     ----------------------------------------------------------
 *          0          S           C             T
 *          1          C          -S            -1/T
 *          2         -S          -C             T
 *          3         -C           S            -1/T
 *     ----------------------------------------------------------
 *
 * Special cases:
 *      Let trig be any of sin, cos, or tan.
 *      trig(+-INF)  is NaN, with signals;
 *      trig(NaN)    is that NaN;
 *
 * Accuracy:
 *      TRIG(x) returns trig(x) nearly rounded
 */
        double sin(double x)
        {
            double y[2], z = 0.0;
            int32_t n, ix;

            /* High word of x. */
            GET_HIGH_WORD(ix, x);

            /* |x| ~< pi/4 */
            ix &= 0x7FFFFFFF;
            if (ix <= 0x3FE921FB) {
                return __kernel_sin(x, z, 0);
            } else if (ix >= 0x7FF00000) {
                /* sin(Inf or NaN) is NaN */
                return x - x;
            } else {
                /* argument reduction needed */
                n = __ieee754_rem_pio2(x, y);
                switch (n & 3) {
                case 0:
                    return __kernel_sin(y[0], y[1], 1);
                case 1:
                    return __kernel_cos(y[0], y[1]);
                case 2:
                    return -__kernel_sin(y[0], y[1], 1);
                default:
                    return -__kernel_cos(y[0], y[1]);
                }
            }
        }

        /* tan(x)
 * Return tangent function of x.
 *
 * kernel function:
 *      __kernel_tan            ... tangent function on [-pi/4,pi/4]
 *      __ieee754_rem_pio2      ... argument reduction routine
 *
 * Method.
 *      Let S,C and T denote the sin, cos and tan respectively on
 *      [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
 *      in [-pi/4 , +pi/4], and let n = k mod 4.
 *      We have
 *
 *          n        sin(x)      cos(x)        tan(x)
 *     ----------------------------------------------------------
 *          0          S           C             T
 *          1          C          -S            -1/T
 *          2         -S          -C             T
 *          3         -C           S            -1/T
 *     ----------------------------------------------------------
 *
 * Special cases:
 *      Let trig be any of sin, cos, or tan.
 *      trig(+-INF)  is NaN, with signals;
 *      trig(NaN)    is that NaN;
 *
 * Accuracy:
 *      TRIG(x) returns trig(x) nearly rounded
 */
        double tan(double x)
        {
            double y[2], z = 0.0;
            int32_t n, ix;

            /* High word of x. */
            GET_HIGH_WORD(ix, x);

            /* |x| ~< pi/4 */
            ix &= 0x7FFFFFFF;
            if (ix <= 0x3FE921FB) {
                return __kernel_tan(x, z, 1);
            } else if (ix >= 0x7FF00000) {
                /* tan(Inf or NaN) is NaN */
                return x - x; /* NaN */
            } else {
                /* argument reduction needed */
                n = __ieee754_rem_pio2(x, y);
                /* 1 -> n even, -1 -> n odd */
                return __kernel_tan(y[0], y[1], 1 - ((n & 1) << 1));
            }
        }

        /*
 * ES6 draft 09-27-13, section 20.2.2.12.
 * Math.cosh
 * Method :
 * mathematically cosh(x) if defined to be (exp(x)+exp(-x))/2
 *      1. Replace x by |x| (cosh(x) = cosh(-x)).
 *      2.
 *                                                      [ exp(x) - 1 ]^2
 *          0        <= x <= ln2/2  :  cosh(x) := 1 + -------------------
 *                                                         2*exp(x)
 *
 *                                                 exp(x) + 1/exp(x)
 *          ln2/2    <= x <= 22     :  cosh(x) := -------------------
 *                                                        2
 *          22       <= x <= lnovft :  cosh(x) := exp(x)/2
 *          lnovft   <= x <= ln2ovft:  cosh(x) := exp(x/2)/2 * exp(x/2)
 *          ln2ovft  <  x           :  cosh(x) := huge*huge (overflow)
 *
 * Special cases:
 *      cosh(x) is |x| if x is +INF, -INF, or NaN.
 *      only cosh(0)=1 is exact for finite x.
 */
        double cosh(double x)
        {
            static const double KCOSH_OVERFLOW = 710.4758600739439;
            static const double one = 1.0, half = 0.5;
            static volatile double huge = 1.0e+300;

            int32_t ix;

            /* High word of |x|. */
            GET_HIGH_WORD(ix, x);
            ix &= 0x7FFFFFFF;

            // |x| in [0,0.5*log2], return 1+expm1(|x|)^2/(2*exp(|x|))
            if (ix < 0x3FD62E43) {
                double t = expm1(fabs(x));
                double w = one + t;
                // For |x| < 2^-55, cosh(x) = 1
                if (ix < 0x3C800000)
                    return w;
                return one + (t * t) / (w + w);
            }

            // |x| in [0.5*log2, 22], return (exp(|x|)+1/exp(|x|)/2
            if (ix < 0x40360000) {
                double t = exp(fabs(x));
                return half * t + half / t;
            }

            // |x| in [22, log(maxdouble)], return half*exp(|x|)
            if (ix < 0x40862E42)
                return half * exp(fabs(x));

            // |x| in [log(maxdouble), overflowthreshold]
            if (fabs(x) <= KCOSH_OVERFLOW) {
                double w = exp(half * fabs(x));
                double t = half * w;
                return t * w;
            }

            /* x is INF or NaN */
            if (ix >= 0x7FF00000)
                return x * x;

            // |x| > overflowthreshold.
            return huge * huge;
        }

        /*
 * ES2019 Draft 2019-01-02 12.6.4
 * Math.pow & Exponentiation Operator
 *
 * Return X raised to the Yth power
 *
 * Method:
 *     Let x =  2   * (1+f)
 *     1. Compute and return log2(x) in two pieces:
 *        log2(x) = w1 + w2,
 *        where w1 has 53-24 = 29 bit trailing zeros.
 *     2. Perform y*log2(x) = n+y' by simulating muti-precision
 *        arithmetic, where |y'|<=0.5.
 *     3. Return x**y = 2**n*exp(y'*log2)
 *
 * Special cases:
 *     1.  (anything) ** 0  is 1
 *     2.  (anything) ** 1  is itself
 *     3.  (anything) ** NAN is NAN
 *     4.  NAN ** (anything except 0) is NAN
 *     5.  +-(|x| > 1) **  +INF is +INF
 *     6.  +-(|x| > 1) **  -INF is +0
 *     7.  +-(|x| < 1) **  +INF is +0
 *     8.  +-(|x| < 1) **  -INF is +INF
 *     9.  +-1         ** +-INF is NAN
 *     10. +0 ** (+anything except 0, NAN)               is +0
 *     11. -0 ** (+anything except 0, NAN, odd integer)  is +0
 *     12. +0 ** (-anything except 0, NAN)               is +INF
 *     13. -0 ** (-anything except 0, NAN, odd integer)  is +INF
 *     14. -0 ** (odd integer) = -( +0 ** (odd integer) )
 *     15. +INF ** (+anything except 0,NAN) is +INF
 *     16. +INF ** (-anything except 0,NAN) is +0
 *     17. -INF ** (anything)  = -0 ** (-anything)
 *     18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
 *     19. (-anything except 0 and inf) ** (non-integer) is NAN
 *
 * Accuracy:
 *      pow(x,y) returns x**y nearly rounded. In particular,
 *      pow(integer, integer) always returns the correct integer provided it is
 *      representable.
 *
 * Constants:
 *     The hexadecimal values are the intended ones for the following
 *     constants. The decimal values may be used, provided that the
 *     compiler will convert from decimal to binary accurately enough
 *     to produce the hexadecimal values shown.
 */

        double pow(double x, double y)
        {
            static const double
                bp[]
                = { 1.0, 1.5 },
                dp_h[] = { 0.0, 5.84962487220764160156e-01 }, // 0x3FE2B803, 0x40000000
                dp_l[] = { 0.0, 1.35003920212974897128e-08 }, // 0x3E4CFDEB, 0x43CFD006
                zero = 0.0, one = 1.0, two = 2.0,
                two53 = 9007199254740992.0, // 0x43400000, 0x00000000
                huge = 1.0e300, tiny = 1.0e-300,
                // poly coefs for (3/2)*(log(x)-2s-2/3*s**3
                L1 = 5.99999999999994648725e-01, // 0x3FE33333, 0x33333303
                L2 = 4.28571428578550184252e-01, // 0x3FDB6DB6, 0xDB6FABFF
                L3 = 3.33333329818377432918e-01, // 0x3FD55555, 0x518F264D
                L4 = 2.72728123808534006489e-01, // 0x3FD17460, 0xA91D4101
                L5 = 2.30660745775561754067e-01, // 0x3FCD864A, 0x93C9DB65
                L6 = 2.06975017800338417784e-01, // 0x3FCA7E28, 0x4A454EEF
                P1 = 1.66666666666666019037e-01, // 0x3FC55555, 0x5555553E
                P2 = -2.77777777770155933842e-03, // 0xBF66C16C, 0x16BEBD93
                P3 = 6.61375632143793436117e-05, // 0x3F11566A, 0xAF25DE2C
                P4 = -1.65339022054652515390e-06, // 0xBEBBBD41, 0xC5D26BF1
                P5 = 4.13813679705723846039e-08, // 0x3E663769, 0x72BEA4D0
                lg2 = 6.93147180559945286227e-01, // 0x3FE62E42, 0xFEFA39EF
                lg2_h = 6.93147182464599609375e-01, // 0x3FE62E43, 0x00000000
                lg2_l = -1.90465429995776804525e-09, // 0xBE205C61, 0x0CA86C39
                ovt = 8.0085662595372944372e-0017, // -(1024-log2(ovfl+.5ulp))
                cp = 9.61796693925975554329e-01, // 0x3FEEC709, 0xDC3A03FD =2/(3ln2)
                cp_h = 9.61796700954437255859e-01, // 0x3FEEC709, 0xE0000000 =(float)cp
                cp_l = -7.02846165095275826516e-09, // 0xBE3E2FE0, 0x145B01F5 =tail cp_h
                ivln2 = 1.44269504088896338700e+00, // 0x3FF71547, 0x652B82FE =1/ln2
                ivln2_h = 1.44269502162933349609e+00, // 0x3FF71547, 0x60000000 =24b 1/ln2
                ivln2_l = 1.92596299112661746887e-08; // 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail

            double z, ax, z_h, z_l, p_h, p_l;
            double y1, t1, t2, r, s, t, u, v, w;
            int i, j, k, yisint, n;
            int hx, hy, ix, iy;
            unsigned lx, ly;

            EXTRACT_WORDS(hx, lx, x);
            EXTRACT_WORDS(hy, ly, y);
            ix = hx & 0x7fffffff;
            iy = hy & 0x7fffffff;

            /* y==zero: x**0 = 1 */
            if ((iy | ly) == 0)
                return one;

            /* +-NaN return x+y */
            if (ix > 0x7ff00000 || ((ix == 0x7ff00000) && (lx != 0)) || iy > 0x7ff00000 || ((iy == 0x7ff00000) && (ly != 0))) {
                return x + y;
            }

            /* determine if y is an odd int when x < 0
   * yisint = 0 ... y is not an integer
   * yisint = 1 ... y is an odd int
   * yisint = 2 ... y is an even int
   */
            yisint = 0;
            if (hx < 0) {
                if (iy >= 0x43400000) {
                    yisint = 2; /* even integer y */
                } else if (iy >= 0x3ff00000) {
                    k = (iy >> 20) - 0x3ff; /* exponent */
                    if (k > 20) {
                        j = ly >> (52 - k);
                        if ((j << (52 - k)) == static_cast<int>(ly))
                            yisint = 2 - (j & 1);
                    } else if (ly == 0) {
                        j = iy >> (20 - k);
                        if ((j << (20 - k)) == iy)
                            yisint = 2 - (j & 1);
                    }
                }
            }

            /* special value of y */
            if (ly == 0) {
                if (iy == 0x7ff00000) { /* y is +-inf */
                    if (((ix - 0x3ff00000) | lx) == 0) {
                        return y - y; /* inf**+-1 is NaN */
                    } else if (ix >= 0x3ff00000) { /* (|x|>1)**+-inf = inf,0 */
                        return (hy >= 0) ? y : zero;
                    } else { /* (|x|<1)**-,+inf = inf,0 */
                        return (hy < 0) ? -y : zero;
                    }
                }
                if (iy == 0x3ff00000) { /* y is  +-1 */
                    if (hy < 0) {
                        return base::Divide(one, x);
                    } else {
                        return x;
                    }
                }
                if (hy == 0x40000000)
                    return x * x; /* y is  2 */
                if (hy == 0x3fe00000) { /* y is  0.5 */
                    if (hx >= 0) { /* x >= +0 */
                        return sqrt(x);
                    }
                }
            }

            ax = fabs(x);
            /* special value of x */
            if (lx == 0) {
                if (ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000) {
                    z = ax; /*x is +-0,+-inf,+-1*/
                    if (hy < 0)
                        z = base::Divide(one, z); /* z = (1/|x|) */
                    if (hx < 0) {
                        if (((ix - 0x3ff00000) | yisint) == 0) {
                            /* (-1)**non-int is NaN */
                            z = std::numeric_limits<double>::signaling_NaN();
                        } else if (yisint == 1) {
                            z = -z; /* (x<0)**odd = -(|x|**odd) */
                        }
                    }
                    return z;
                }
            }

            n = (hx >> 31) + 1;

            /* (x<0)**(non-int) is NaN */
            if ((n | yisint) == 0) {
                return std::numeric_limits<double>::signaling_NaN();
            }

            s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
            if ((n | (yisint - 1)) == 0)
                s = -one; /* (-ve)**(odd int) */

            /* |y| is huge */
            if (iy > 0x41e00000) { /* if |y| > 2**31 */
                if (iy > 0x43f00000) { /* if |y| > 2**64, must o/uflow */
                    if (ix <= 0x3fefffff)
                        return (hy < 0) ? huge * huge : tiny * tiny;
                    if (ix >= 0x3ff00000)
                        return (hy > 0) ? huge * huge : tiny * tiny;
                }
                /* over/underflow if x is not close to one */
                if (ix < 0x3fefffff)
                    return (hy < 0) ? s * huge * huge : s * tiny * tiny;
                if (ix > 0x3ff00000)
                    return (hy > 0) ? s * huge * huge : s * tiny * tiny;
                /* now |1-x| is tiny <= 2**-20, suffice to compute
       log(x) by x-x^2/2+x^3/3-x^4/4 */
                t = ax - one; /* t has 20 trailing zeros */
                w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25));
                u = ivln2_h * t; /* ivln2_h has 21 sig. bits */
                v = t * ivln2_l - w * ivln2;
                t1 = u + v;
                SET_LOW_WORD(t1, 0);
                t2 = v - (t1 - u);
            } else {
                double ss, s2, s_h, s_l, t_h, t_l;
                n = 0;
                /* take care subnormal number */
                if (ix < 0x00100000) {
                    ax *= two53;
                    n -= 53;
                    GET_HIGH_WORD(ix, ax);
                }
                n += ((ix) >> 20) - 0x3ff;
                j = ix & 0x000fffff;
                /* determine interval */
                ix = j | 0x3ff00000; /* normalize ix */
                if (j <= 0x3988E) {
                    k = 0; /* |x|<sqrt(3/2) */
                } else if (j < 0xBB67A) {
                    k = 1; /* |x|<sqrt(3)   */
                } else {
                    k = 0;
                    n += 1;
                    ix -= 0x00100000;
                }
                SET_HIGH_WORD(ax, ix);

                /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
                u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
                v = base::Divide(one, ax + bp[k]);
                ss = u * v;
                s_h = ss;
                SET_LOW_WORD(s_h, 0);
                /* t_h=ax+bp[k] High */
                t_h = zero;
                SET_HIGH_WORD(t_h, ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18));
                t_l = ax - (t_h - bp[k]);
                s_l = v * ((u - s_h * t_h) - s_h * t_l);
                /* compute log(ax) */
                s2 = ss * ss;
                r = s2 * s2 * (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6)))));
                r += s_l * (s_h + ss);
                s2 = s_h * s_h;
                t_h = 3.0 + s2 + r;
                SET_LOW_WORD(t_h, 0);
                t_l = r - ((t_h - 3.0) - s2);
                /* u+v = ss*(1+...) */
                u = s_h * t_h;
                v = s_l * t_h + t_l * ss;
                /* 2/(3log2)*(ss+...) */
                p_h = u + v;
                SET_LOW_WORD(p_h, 0);
                p_l = v - (p_h - u);
                z_h = cp_h * p_h; /* cp_h+cp_l = 2/(3*log2) */
                z_l = cp_l * p_h + p_l * cp + dp_l[k];
                /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
                t = static_cast<double>(n);
                t1 = (((z_h + z_l) + dp_h[k]) + t);
                SET_LOW_WORD(t1, 0);
                t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
            }

            /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
            y1 = y;
            SET_LOW_WORD(y1, 0);
            p_l = (y - y1) * t1 + y * t2;
            p_h = y1 * t1;
            z = p_l + p_h;
            EXTRACT_WORDS(j, i, z);
            if (j >= 0x40900000) { /* z >= 1024 */
                if (((j - 0x40900000) | i) != 0) { /* if z > 1024 */
                    return s * huge * huge; /* overflow */
                } else {
                    if (p_l + ovt > z - p_h)
                        return s * huge * huge; /* overflow */
                }
            } else if ((j & 0x7fffffff) >= 0x4090cc00) { /* z <= -1075 */
                if (((j - 0xc090cc00) | i) != 0) { /* z < -1075 */
                    return s * tiny * tiny; /* underflow */
                } else {
                    if (p_l <= z - p_h)
                        return s * tiny * tiny; /* underflow */
                }
            }
            /*
   * compute 2**(p_h+p_l)
   */
            i = j & 0x7fffffff;
            k = (i >> 20) - 0x3ff;
            n = 0;
            if (i > 0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
                n = j + (0x00100000 >> (k + 1));
                k = ((n & 0x7fffffff) >> 20) - 0x3ff; /* new k for n */
                t = zero;
                SET_HIGH_WORD(t, n & ~(0x000fffff >> k));
                n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
                if (j < 0)
                    n = -n;
                p_h -= t;
            }
            t = p_l + p_h;
            SET_LOW_WORD(t, 0);
            u = t * lg2_h;
            v = (p_l - (t - p_h)) * lg2 + t * lg2_l;
            z = u + v;
            w = v - (z - u);
            t = z * z;
            t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
            r = base::Divide(z * t1, (t1 - two) - (w + z * w));
            z = one - (r - z);
            GET_HIGH_WORD(j, z);
            j += static_cast<int>(static_cast<uint32_t>(n) << 20);
            if ((j >> 20) <= 0) {
                z = scalbn(z, n); /* subnormal output */
            } else {
                int tmp;
                GET_HIGH_WORD(tmp, z);
                SET_HIGH_WORD(z, tmp + static_cast<int>(static_cast<uint32_t>(n) << 20));
            }
            return s * z;
        }

        /*
 * ES6 draft 09-27-13, section 20.2.2.30.
 * Math.sinh
 * Method :
 * mathematically sinh(x) if defined to be (exp(x)-exp(-x))/2
 *      1. Replace x by |x| (sinh(-x) = -sinh(x)).
 *      2.
 *                                                  E + E/(E+1)
 *          0        <= x <= 22     :  sinh(x) := --------------, E=expm1(x)
 *                                                      2
 *
 *          22       <= x <= lnovft :  sinh(x) := exp(x)/2
 *          lnovft   <= x <= ln2ovft:  sinh(x) := exp(x/2)/2 * exp(x/2)
 *          ln2ovft  <  x           :  sinh(x) := x*shuge (overflow)
 *
 * Special cases:
 *      sinh(x) is |x| if x is +Infinity, -Infinity, or NaN.
 *      only sinh(0)=0 is exact for finite x.
 */
        double sinh(double x)
        {
            static const double KSINH_OVERFLOW = 710.4758600739439,
                                TWO_M28 = 3.725290298461914e-9, // 2^-28, empty lower half
                LOG_MAXD = 709.7822265625; // 0x40862E42 00000000, empty lower half
            static const double shuge = 1.0e307;

            double h = (x < 0) ? -0.5 : 0.5;
            // |x| in [0, 22]. return sign(x)*0.5*(E+E/(E+1))
            double ax = fabs(x);
            if (ax < 22) {
                // For |x| < 2^-28, sinh(x) = x
                if (ax < TWO_M28)
                    return x;
                double t = expm1(ax);
                if (ax < 1) {
                    return h * (2 * t - t * t / (t + 1));
                }
                return h * (t + t / (t + 1));
            }
            // |x| in [22, log(maxdouble)], return 0.5 * exp(|x|)
            if (ax < LOG_MAXD)
                return h * exp(ax);
            // |x| in [log(maxdouble), overflowthreshold]
            // overflowthreshold = 710.4758600739426
            if (ax <= KSINH_OVERFLOW) {
                double w = exp(0.5 * ax);
                double t = h * w;
                return t * w;
            }
            // |x| > overflowthreshold or is NaN.
            // Return Infinity of the appropriate sign or NaN.
            return x * shuge;
        }

        /* Tanh(x)
 * Return the Hyperbolic Tangent of x
 *
 * Method :
 *                                 x    -x
 *                                e  - e
 *  0. tanh(x) is defined to be -----------
 *                                 x    -x
 *                                e  + e
 *  1. reduce x to non-negative by tanh(-x) = -tanh(x).
 *  2.  0      <= x <  2**-28 : tanh(x) := x with inexact if x != 0
 *                                          -t
 *      2**-28 <= x <  1      : tanh(x) := -----; t = expm1(-2x)
 *                                         t + 2
 *                                               2
 *      1      <= x <  22     : tanh(x) := 1 - -----; t = expm1(2x)
 *                                             t + 2
 *      22     <= x <= INF    : tanh(x) := 1.
 *
 * Special cases:
 *      tanh(NaN) is NaN;
 *      only tanh(0)=0 is exact for finite argument.
 */
        double tanh(double x)
        {
            static const volatile double tiny = 1.0e-300;
            static const double one = 1.0, two = 2.0, huge = 1.0e300;
            double t, z;
            int32_t jx, ix;

            GET_HIGH_WORD(jx, x);
            ix = jx & 0x7FFFFFFF;

            /* x is INF or NaN */
            if (ix >= 0x7FF00000) {
                if (jx >= 0)
                    return one / x + one; /* tanh(+-inf)=+-1 */
                else
                    return one / x - one; /* tanh(NaN) = NaN */
            }

            /* |x| < 22 */
            if (ix < 0x40360000) { /* |x|<22 */
                if (ix < 0x3E300000) { /* |x|<2**-28 */
                    if (huge + x > one)
                        return x; /* tanh(tiny) = tiny with inexact */
                }
                if (ix >= 0x3FF00000) { /* |x|>=1  */
                    t = expm1(two * fabs(x));
                    z = one - two / (t + two);
                } else {
                    t = expm1(-two * fabs(x));
                    z = -t / (t + two);
                }
                /* |x| >= 22, return +-1 */
            } else {
                z = one - tiny; /* raise inexact flag */
            }
            return (jx >= 0) ? z : -z;
        }

#undef EXTRACT_WORDS
#undef EXTRACT_WORD64
#undef GET_HIGH_WORD
#undef GET_LOW_WORD
#undef INSERT_WORDS
#undef INSERT_WORD64
#undef SET_HIGH_WORD
#undef SET_LOW_WORD
#undef STRICT_ASSIGN

    } // namespace ieee754
} // namespace base
} // namespace v8
